Why is this not the solution?

[PredatoryCatgirl]

A bidding system would be the same as other players choosing who may and may not buy a house, with those rich enough able to guarantee a purchase by others of their choosing.

Why would anyone do such a thing, you ask?

If I had enough gil, I would certainly surround myself with friends and family, or those who owe me a great deal...

Or I could simply deny someone I don't like from getting a home.

Absolute power corrupts absolutely. You'd have realties introduced into the game. Or worse, multiple HMAs.

This system is already in place unofficially, and that doesn't happen. There are people who buy and sell FCs with housing attached to them. All this would change is make that money go into the void rather than the pockets of another player, thereby being the gil sink we need.