I assume that a genre-specific toolbox would not suffer from the same issues.
I feel like this is a mistaken assumption because the Urban Fantasy genre is so extremely broad. And it's been attempted - Unisystem was basically the "Urban Fantasy genre-specific RPG" back in the 1990s.
Just in this thread, we're talking about everything from Shadowrun, to Dresden, to City of Mists, to Urban Shadows, to World of Darkness and stranger games too.
There's no system that could accomodate all those games and their associated worlds/settings/magic/etc. comfortably without it being broadly generic and having the exact issue
@Umbran described.
AFMBE is a poor example for two reasons:
1) It is a generic RPG, just with specific rules in a single book - specifically, it's Unisystem. Which I've already mentioned was basically a generic RPG focused on Urban Fantasy-type stuff. It's not an zombie-specific RPG. It's just a subset of the Unisystem rules for running zombie-oriented games.
2) AFMBE is pretty generic-feeling and whilst the advice about designing zombie settings and campaigns is good, it could be applied to any number of RPGs, and AFMBE/Unisystem is not, in fact, a great system for running zombie campaigns, I would argue (admittedly based only on having played it a little - but I was a huge fan of zombies back then).
So if we went with AFMBE as the example of way to do this, all you would do is take, say, Cortex Prime, or Cypher, or Fate, or whatever, just like they did with Unisystem, and then just build out an absolute ton of
examples using that generic system. That gives you a toolbox. But the more flexible it is, the less flavour and distinctiveness it is likely to have.