...how would having to pay rent make apartments more appealing? Even if it comes with a reduced initial cost.

How would the rent even be paid? Would you have to walk up to an NPC every week to pay? As far as I know there is no system (yet) that would just automatically take gil from your inventory - if my assumption here is correct and they would have to implement a system like that in the first place, I'd think it a waste of ressources. And if such a system could/would work: Would it be able to take gil out of your retainers pockets? Or do you need to make sure you always have the needed amount on your person? (Personally, I keep most of my gil on my retainers)

What purpose would it serve to evict people from their apartments? On the majority of worlds, there are enough apartments for anyone who wants one (Balmung might be an exception, never been there). Auto-demolition for houses is a thing because there arent enough for everyone, so they want to make sure that only people who use their house get to keep it. But apartments face no such shortage, so theres not need for eviction or auto-demolition.

All in all... what a terrible idea that doesnt seem to serve any purpose (more realism?).