Have a massive empire and then lose it resulting in a capital city with surplus housing?

That was Berlin several decades ago. The gap has closed now and it's just another city.

Well, Berlin's history was a bit more complicated. Between the 'losing the empire' bit and the 'surplus housing' was a whole generation or two of the city being divided and one part living under siege and the other under real world socialism.

But you are right, that surplus housing is no longer a thing in Berlin.

I don't know about restricting rent but they certainly had a progressive tax on high rents, a 1% income tax to fund housing, and then used that to build a ton of high quality social housing with balconies and amenities for all. I also like to think it helped that they slapped a big sign on each to proudly let people know it was their tax dollars at work.

I didn't stay in social housing while I living there but I never once heard people complain about it. They basically just didn't think much about it at all and felt it was a good system and then would ask me why the US only makes it for poor people.