I am Weber. I prefer the comfort of my home kitchen and the warmth of fresh bread... but for my country... I will do what I must. Use my skills with tomes and staves as you see fit.
I think FE’s core mechanics (permadeath especially) are naturally designed to foster the type of story creation that you’re describing, so you’ll always get at least some amount of that effect no matter what as long as those traits are still a part of the game.
Despite that, I’m not sure if it’s enough to rely solely on those mechanics to generate a cast’s personal stories, either. Rather, I want to say that style of characterization works most smoothly when the game is designed around it—i:e if the game is willing to commit to setting itself up in a way that eases the player’s process of story generation. As for the mainline games, I want to say that the Kaga games are generally pretty good at this, thanks to the way they lay their story foundation and frame their casts as a part of the established world. Even units like Ralph or Warren or the worse-written subs still feel like they were included in the story for a reason, which makes it easier to write your own stories in your head about them.
In terms of existing hacks, Iron Emblem accomplishes this effect pretty well (minus the characters having unique appearances and what not) albeit for a somewhat different reason. The story and lore being as barebones as it is goes a long way in framing it as more of a generative story sandbox than a traditional story, and I think that the writing being consistently nonexistent is an important part of what makes this work. Were this minimal dialogue style not universal for all the characters in the game, I don’t think it would have been half as effective.
Speaking personally, as someone whose first mainline FE game was FE3, this kind of experience will always hold a special place in my heart. But while I think it’s a vital part of FE’s identity, I’m not confident that it can carry a cast of characters on its own without being handled mindfully, especially in the realm of hacks where I’ve come to expect a generally higher ceiling of narrative refinement than in vanilla.
TL:DR just like anything, I feel it can be made to work as long as the context is right, but when handled carelessly it could easily come off poorly.