One of my favorite things about 5e is its simplicity, but one of my favorite things about 3.x was the capacity for new rules system (I never had be worried about power creep back then, because WotC wasn't either

).
I don't think that 5E is less open to new rules systems than 3E. Just the opposite, in fact. I've tinkered with every edition of D&D since BECMI and generally liked the results -- until 3E. Really tinkering with the core of 3E tended to end badly because it was so tightly coupled.
5E's simplicity is also a strength. You just can't think of it in the same terms as 3E; it's much more like 1E/2E. Just know the bounds of what you can change. You
can slap on kits without totally breaking the game (whether you want to is a different question). You can add descriptive critical hits or critical fumbles. Ignore feats or add feat trees (bad idea, IMO). You could even make Background selection meatier by adding in benefits at, say, every five levels or make it possible to "replace" your background by roleplaying through significant changes. You could even pull Fate aspects into 5E, wholesale, using advantage/disadvantage for the mechanical impacts.
To change the sub-classing system though, means
changing the sub-classing system. That's pretty hard-wired into the game. The impact is that you'd end up having to rebalance the whole thing so that all classes get sub-class abilities at the same levels and they're getting roughly the same level of oomph from their sub-class vs. the parent class. It's doable, but passes my ROI cutoff by a fair margin. It also means that your "universal" sub-class is something that needs to be locked in by about 3rd level. My gut says that the best use of universal sub-classes is to represent some RP aspect that you want to have grant a mechanical benefit, say Harper Agent, which could easily come at 10th level or so. I'd either use a feat for the couple of perks that come with that or just RP it.
On the other hand, if being locked it at 3rd level is okay, then using a background is probably fine, too. If you want something meatier than the existing backgrounds, replace the standard Ability Score Increase (4th, 8th, etc.) with background benefits. I'd want to do it for all backgrounds, just for consistency, but it would be a fine mechanic. Alternatively, especially if keeping the feat mechanic is important to your group, just add in a couple of feats with certain backgrounds as prerequisites.