*shrug* I think you're getting overly caught up on the use of the word "background" (and the downtime rules are as terrible in 5E as they've been in every edition). Call it "profession", whatever you want. The name isn't important. The "background" just provides a base. You were a criminal, do you seek redemption? To force a criminal empire? To skirt the law for fun and for profit?
That's addressed by Traits, Ideals, Bonds, and Flaws.
There, just like 5E's classes, each background now gains a "subclass" that you can advance into based on the direction you'd like to take your character role-play wise.
If you need crunch to encourage you to play to your Traits, Ideals, Bonds, and Flaws, you're missing the point of Traits, Ideals, Bonds, and Flaws.
I'm saying D&D lacks a solid framework to promote RP friendly choices, often forcing players to choose between creatively interesting choices and mechanically beneficial ones. Having a secondary system for RP advancement only would go a long way to alleviating that.
Oh, I'm sure - for those who don't understand the concept of Traits, Ideals, Bonds, and Flaws.

People who understand it and buy into it are already doing all of that, exactly BTB RAW.
It ain't broke.
But plenty of d20 games that were otherwise very much like DnD made it work. The D20 modern game had the cha, int, and wisdom based heroes. Star Wars d20and saga edition had the Noble. Conan d20 had a Noble and a Temptress class. Fantasycraft has an Explorer, Sage, Keeper, and more. Some contribute more in a fight than others, but all do a good job of letting you play a character less focused on stabbing stuff in a fight.
This right here encapsulates why discussions about optimization are so annoying: Optimization is invariably focused solely on in-combat mechanics. As though a CHA-focused intrigue character can't possibly be of any use.
I had a table like that once. They gave one player endless grief because he wanted to play well-rounded characters, not just damage-dealing murderhobos. He was about to quit the game over it. That made me think about my approach as DM, as well as their behavior and focus. Turns out I was rewarding that by making the game an endless stream of wilderness and dungeon combat experiences. So I threw them into a major settlement where the excellent killers were certainly
not optimized; in fact, they were virtually useless.
They burned that city down and went back into the wilderness.
So I did it again. And again.
I learned that combat-focused optimization is fine
in its place. There should also be room in the game for those who don't optimize, or who choose to optimize in different ways.
Cheers,
Bob
www.r-p-davis.com