Why do we still have enmity?

[Reynhart]

But, somehow, swapping to that DPS combo over the enmity combo won't cause you to generate such a low amount of enmity that you start to lose some of your lead... Which of course wouldn't then mean you just spammed DPS combos 24/7 and Provoked on CD because reasons?

The lead is mainly counted in Floors. If you don't have enough time to finish a floor before using Provoke, the enmity you'd have gained by your enmity combo is wasted. If, during that time, one of the DPS gain one Floor, you just can't do anything about it, since he would have gained that Floor anyway.
As for spamming Provoke on CD, is also depend on the rest of the toolkit. Of course, if enmity becomes a proper mechanic, we can expect tanks to have several skills to manage it. Also, if Provoke became available when you're on the verge of gaining a Floor, you'd obviously delay it after you've gained that floor to gain another one right away. Exactly like DRG wouldn't use Life Surge if the next weapon skill is True Thrust.

To say nothing about how this system works with Tank swaps? Who's going to get aggro? What happens if both tanks are simply spamming Provoke on CD to generate "Floors"?

It's not always strategic for the OT to gain too much floors, since it would only prevent him from gaining the full buff. As for tank swaps, Shirk would already help a lot and would basically transfer floors to someone else. You could also have skills that spend floors for other uses, to purposefully reduce your enmity, and let someone go above you. But you'd probably would have to do some stuff to facilitate the swap instead of simply letting the other Provoke. On a sidenote, I personally think that if so much fight require a tank swap, it's also because that's one of the only mechanics related to enmity.

Or will this literally be just another completely unrelated mechanic while enmity continues to work as current and be a non-factor?

Considering that the basic effect of enmity is "who's attacked", and that the target would still be the one with the highest floor, this would still be counted as "enmity".