Blizzard has made it clear that artifact weapons are going to be a Legion-only thing. And honestly, that makes sense. They've really up-ended one of the core systems to the game, though I've got to say that they did it in a fairly smooth way.
As someone whose main uses two different pieces of equipment in his hands (a melee weapon and a shield) I can definitely say that it pleases my OCD tendencies that both items are always at the same item level (granted, it's less pleasing that I still have a blue-quality relic in the Arcane slot, even though I know that relics are actually only a third as important as any other piece of gear.)
We're still early in this expansion - an expansion that Blizzard claims is planned to take longer than previous expansions were planned to. 7.1's content list seems a good indicator of this philosophy - we're getting a new raid and dungeon and new world content, but we're still not actually going beyond the first raiding tier - in fact, despite the fact that the Nighthold raid is in-game as part of 7.0, it won't actually be accessible until after 7.1.
There's a real build-up to Nighthold, which is not even planned to be the expansion's final raid (my guess is Tomb of Sargeras, unless they surprise us and send us to Argus in this expansion rather than saving it for a later one.)
The point is that we're not going to be ditching our artifact weapons soon.
I doubt that anyone has been able to completely fill out their artifact traits yet - even if you've done every single world quest and run dungeons non-stop to maximize your AP acquisition, the exponential nature of the costs of artifact traits means that artifact knowledge is really required to finish them off.
My main hit 110 either the day after or two days after the expansion launched, and I've had his artifact research ticking non-stop during that time. He's currently got an artifact knowledge level of I think 7 (it's an additional 375%.) He's certainly well on the way to filling out his artifact - there are six minor traits to collect before he gets his final major trait, and then only a few more to get after that.
The point here is that players will have their artifacts filled out within the next couple months, possibly before 7.1 comes out.
Now obviously for many of us this will mean that we can now start pouring AP into off-spec artifacts (given that I never play Holy, I've decided to wait until I have the absolute highest level of artifact knowledge before I put a single point into the Silver Hand, just to see how quickly I can fill out its tree.)
Now there are "prestige" traits that unlock after getting the main 34 traits unlocked. These go up to I think 50 and give a minuscule amount of damage, healing, and health, but are there for when you have nothing else to spend AP on. I imagine these traits cost so much that only the truly insane will ever fill out all 50 of them.
For the remainder of Legion, I imagine that the main way to build on the system will be to release new cosmetic skins for them. There's already another set in the game files that is not accessible yet in-game (I'm particularly hopeful that I can get the "Crest of Holy Fire" on my main, which both looks cool and also replaces the sword with a flail.) My hope is that we'll see more artifact models in future patches as well.
Now, regarding the future beyond Legion, I think there are a few things to consider.
We'll presumably be getting some changes to each spec again - this is the constant with all new expansions. Some artifact traits, I think, are far more central than others, and I wonder which might be incorporated into the spec's ordinary spells. I think the newer and more changed specs are more likely to keep theirs. For instance, the Protection Paladin artifact ability is a 1-minute cooldown that does damage to nearby enemies and reduces the damage they do to the Paladin. Certainly it's a welcome ability, but if it were taken away I doubt that the spec would fall apart.
On the other hand, the Vengeance Demon Hunter artifact ability hits the target for some fire damage and causes them to pop off a guaranteed five soul fragments while they burn. This really helps the Demon Hunter get the healing they need to survive without being at the whim of the RNG gods. A friend who plays a Balance Druid was worried that the rotation had become kind of flat and uninteresting in the pre-patch, but upon getting the artifact ability, he said that it gave the rotation structure.
I imagine that Blizzard will have to go through each of these and see what feature should stay and which should go.
Obviously, the first weapon they give us in the next expansion is going to have to be incredibly powerful to get even someone who just leveled up through the Broken Isles to ditch their artifact. I actually suspect they might even have a quest very early on that forces us to give up our artifact weapons so that people will move on without them.
Cosmetically, collecting all these skins for our weapons should have a long-term benefit. I'm hoping that any skins we've unlocked will be available for transmog. I don't know if they'll be able to keep the spec-requirements (that might involve some tricky coding,) but given how much effort it is to get some of these appearances, I do hope Blizzard is thinking long-term about them.