Why is this not the solution?

[ItMe]

So why aren't we using a bidding system when a house goes up on the market?

Because that would make housing less accessible.
Instead of an open plot being FFA it would go to the wealthiest player.
Housing would be 100% unattainable for new players, or heck probably even people that started in the last expac or two. There are players with capped gil counters and if it comes to a bidding war only the richest players will enjoy housing.

If the idea is to act as a gil drain, this might accomplish that (though I doubt players will be happy spending millions or billions of gil on something that was thousands before).

If the idea is to "fix housing," it's kinda just replacing the competition problem with another problem, especially considering how few plots are ever open this is most definitely not the way to go.