The Bladesinger and the Eldritch Knight are two sides of the same coin - one is the Fighter who gets a bit of wizardry and the other is a Wizard who becomes a martial class. While I love my EK dearly (and I'm champing at the bit for a chance to play one again - ideally bringing my old character Jax back because our party was ostensibly temporarily in the Nine Hells but the campaign fell apart before we could get out, and I don't love that as a fate for him) I'll also say that I'm almost certain a Bladesinger is the more powerful option, simply because Wizards are, well, super-powerful.
Now, part of that power is that you don't actually have to use all that martial stuff. Literally any Wizard of high-enough level could get their hands on Conjure Minor Elementals and Scorching Ray and do just absurd amounts of damage, scaling remarkably well each time you get access to higher-level spell slots (you could even multiclass to Sorcerer if you have enough Charisma to get Innate Sorcery because your spell slot progression will continue even if you can't pick higher-level spells).
The math is real easy for that - at 10th level, you have 5th level spell slots, so you'd just cast CME at 5th level on turn 1, and then for the rest of combat, get within 15 feet of your targets and burn your other high-level slots on Scorching Ray (maybe reserving your 5th level to recast if you lose concentration). If you have a +9 to hit with your spells, you'll land it on a 5 or higher against a wyvern. That's (3d8+2d6)x5 if you're doing a 4th level Scorching Ray. Each one lands for 20.5, and so if you've got an 80% chance to hit, and then 5% chance to crit, each one's landing for 20.5x85%, or 17.425, so a 4th level spell slot gets you freaking 87 damage per turn.
Even when we account for the first turn requiring set-up, that's still an average of 43.5 per turn in a two-round combat, and it jumps up to 58 over three turns, making it far better than any of these other builds (though the Eldritch Knight, pulling off far more shenanigans, does comparable stuff, and that was even with the Staff-and-Board build, meaning that a Great Weapon Master build would probably do better.
Yes, CME is kind of a busted spell, and even with its nerf in errata, it's still among the most powerful in the game when used like this. (Scorching Ray is fine, but surely wasn't designed with the assumption you'd be adding damage to each attack).
But what if we wanted something more sustainable?
Here's the limitation we'll set: we are not going to use more than one spell slot. We're going to make use of our Bladesong to make melee attacks.
In other words, we're going to pick some decent concentration spell and then go to town.
Let's start building:
Stats:
Funny story: when I rolled stats on the Wizard I've now been playing almost five years, I was tempted to go Bladesinger, because I rolled so well, starting with an 18 in Int, +3 to Con, and +2 to Dex. But I had already come up with a story that very firmly made him an Order of Scribes Wizard (he was an apprentice archivist at the Cobalt Soul in Port Damali, and during the campaign he's been made an Expositor).
The new Bladesinger, though, is a little less stat-dependent, due in large part to the fact that your Bladesong now allows you to use Intelligence for your attacks.
Having decent Dexterity is still not a bad thing - it'll still increase your AC, and indeed, given that you can no longer wear any armor, rather than getting access to Light Armor, so that Dex is going to be slightly more precious. That said, Bladesong will also give you AC equal to your Intelligence modifier - we still benefit from Dex, but as long as we're decent, we can focus our attention on just boosting Int as far as it will go (indeed, we also get that for Concentration saves, meaning we're truly Int all the way).
Gear:
The interesting thing is that while the SCAG/Tasha's Bladesinger would pretty much always go rapier, as it's the highest-damage finesse weapon, you actually have a lot of freedom now - you can go with a Warhammer, even, as long as you're not wielding it in its Versatile mode. Basically, a d8 weapon is going to be our assumption, but whether it's a Longsword, Rapier, Flail, Harpoon, or whatever is kind of irrelevant. We don't get Weapon Mastery, so it's really up to you (and up to what magic weapons your DM is handing out). Now, you could pick some alternatives. If you get Polearm Master, a Quarterstaff or Spear are both solid options. And if you want a little bit of reach, you can take a Whip, which does step you down two damage dice, but the reach can be great if you want to attack outside of the reach of most creatures.
And no armor, so that's simple.
Feats:
I'd be tempted to grab some martial feats - Polearm Master would actually work great if we wanted to fight with a Spear or Quarterstaff (the Pole Strike making up for the lower damage die) but I also think we need to race to a 20 Int here. As such, what I might do is just grab War Caster at level 4 (assuming we've gotten a 17 Int from our background) and then just ASI at 8 to ensure we're capped on it. That actually leaves us with the freedom to pursue non-Int General Feats at levels 12 and 16 if we get to those levels without guilt.
For Origins Feats, of course, I tend to ignore these. Savage Attacker could boost us slightly, but I generally don't find it worth it, and would prefer Lucky, Tough, or Alert.
Spells:
While we're not spending more than one spell slot, we will be using a Cantrip, which we can wave into our Multiattack. The cantrips will of course be either Green-Flame Blade or Booming Blade, the former if there's an opportunity for cleave (or we're fighting something that is vulnerable to fire or otherwise harmed more by it like a Troll) and the latter in single-target situations.
Now, what of our concentration spell?
Well, CME is still a very solid choice. At 5th level, we'd be adding 3d8 damage to each of our attacks, which could mean 6d8 total. If that's what we're getting, are there any spells that can beat it?
Wizards have a lot of spells, so this is a challenging question. We're going to ignore spells that hinder a specific foe, as we'd prefer to just have all this active to attack as many targets as we like, even if we're assuming we're just hitting a Wyvern in a white room. We also don't want one that uses our action, as we want to take advantage of our Extra Attack, so nothing like Dragon's Breath.
Haste would net us an extra attack each turn, but our attacks themselves don't actually deal a ton of damage. We're doing like 1d8+5 per attack, so 9.5 versus the 3d8 we'd be getting from CME, or 13.5, or really 6d8 or 27 given two attacks. Now, if we had something like a Vicious Weapon, this would start to look more appealing, but even then it would only give us one extra attack that deals 1d8+2d6+5, or 16.5, which still isn't matching that.
What about Summon Spells? One among this category that I think has gotten a buff thanks to the fact that fewer monsters have resistance/immunity to nonmagical kinetic damage is Summon Elemental. While there's not as much utility to them as something like an Undead spirit, they hit hard. If we cast this at 4th level, we get two attacks from them for 1d10+8, which winds up being 13.5 damage, and that means that we're actually doing about as much with them as we would with CME at 5th level (it's slightly less because their crits don't hit as hard). Some other advantages in this specific situation are that an Earth Elemental is resistant to piercing and slashing damage, and they're all immune to poison, which makes it harder for the Wyvern to actually kill them.
And you know what, CME gets used so much. This is more fun. Summon Elemental it is! (And this leaves us a nice 5th level slot if we need it later on.)
Actually, before we move on, I wanted to mention a fun one that my best friend's Bard sometimes uses in our Ravnica game - Yolande's Regal Presence. This lets you knock back foes and make them prone if they enter an emanation around you, or if your emanation enters their space. It deals 4d6 psychic damage on a failed wisdom save, or half as much as a success. The reason I'd consider this is to take advantage of Booming Blade, similar to what we did with the Eldritch Knight's use of the Push mastery. But the sequence here isn't quite right - we'd need to bounce them and then run up to them to take advantage if we knocked them prone.
Gameplan:
Pretty simple: if possible, we have the Elemental up (we're going Earth - while it'd be good to have a flying one to chase the Wyvern down, those lesser dragons are so fast that our Elemental wouldn't be able to catch up anyway, so it doesn't really matter). If not, we spend our first turn casting that. Then, as a bonus action, we activate our Bladesong. It honestly might not be a terrible idea for the Earth Elemental to try to grapple the Wyvern just to keep it in place, though your DM might be very strict about whether Extra Attack and Multiattack are the same thing, and if they can use one attack to grapple and a second to make a Slam.
The Elemental makes its attacks and then, turn two, we step in and start attacking as well.
And that's about it. I'll be honest, I'm not predicting this to do amazing damage, but we'll see. This is a resource-conservative build - we know we can go hog wild on them and do insane damage when we want, but this is for serious dungeon crawling.
Mathing It Out:
Both we and our Elemental have a +9 to hit, which means an 80% hit chance.
The Elemental hits for 1d10+8, or 13.5, and an extra 5.5 on a crit. That's 10.8 plus .275, so 11.075 per attack. That then becomes 22.15 with two attacks.
Turn two, we're going to make two attacks, but the first is Booming Blade (or Green-Flame Blade).
Thus, this one is going to hit for 2d8+5, or 14, adding 9 on a crit, so 14x80% is 11.2 and 9x5% is .45, so we get 11.65 damage here.
Then, our second attack is just a normal attack for 1d8+5, so 9.5 on a hit and adding 4.5 on a crit. 9.5x80% is 7.6 and 4.5x5% is .225, so we get 7.825, or 7.83 to round off.
Thus, by turn two, we get a total of 41.63, or rounded off to 41.6.
And you know what, I think that's actually pretty damned respectable. It's not blowing the socks off of our other builds, but this is doing it all with a single spell slot that isn't even our highest-level slot.
There are also some nice defensive benefits here:
The Earth Elemental, for one, is going to be resistant or immune to all of the damage the Wyvern puts out. The Wyvern's most deadly attack is its tail stinger, but the Earth Elemental will take only 5 damage on average from it (when it normally does 35 total).
As for us, if we assume a +2 to Dex, we can cast Mage Armor, which, along with Bladesong, will put our AC at 20. Then, the Shield spell can push that to 25. Get our hands on some Bracers of Defense, and you're matching my super-tanky Eldritch Knight. Then, if we managed a +3 to Con, we'll get a +8 to concentration saves. On top of this, if we do get hit anyway, Song of Defense will allow us to spend spell slots to prevent 5 damage per spell level, which also means lowering the concentration save DC by 10 per spell level (of course still a minimum of 10). In other words, if we took the full burst of an Ancient Green Dragon's breath weapon (for other dragons we'd just use Absorb Elements,) which does 77 damage on average, we could expend a 5th level spell slot to drop that by 25. The 32 remaining damage would call for a concentration save with a DC of 16, which we would get on an 8 or higher, but with advantage, that 65% chance becomes an 87.75% chance to maintain concentration (our elemental would be fine). I mean, for sure if it was an ancient dragon and we were level 10, we'd want to spend our turn getting the hell out of there as quickly as we could, but still. Not bad.