Yes there are restrictions on supply but you need to look at why for the things you mention. Zoning doesn't happen in a vacuum. It's implemented by politicians. Why? What forces are pushing for that? Is it the NIMBYs? Maybe. But money talks in politics. So what political interests are pushing for restrictive zoning? Why?
The real problem here is that housing became financialized (like every aspect of our lives). People see their $300k house go up in value to $600k and think they've made money. But have they? You still need a place to live. If every house that cost $300k now costs $600k, you still only have one house unit of wealth. So what have you really gained?
One extension of this is you decide you need to own multiple properties. You need to become a landlord. For many, this is their retirement. But what is being a landlord? It's withholding a basic human need (ie shelter) for profit. it's choosing to let people die on the streets if it makes you money.
So this system only really benefits the truly wealthy to accumulate wealth by extracting wealth through rent at the threat of violence. Eviction is violence. Witholding shelter is violence.
Now this might be fine if the government provided a baseline for shelter through what is most commonly referred to as "social housing". Americans, in particular, who hear this think "projects" or "ghettoes" (ie Section 8 Housing) but look at a city like Vienna where ~60% (IIRC) of the housing is owned by the government [1].
But we don't have such a aystem in the US because it would reduce profits. It would reduce the extraction of wealth from workers. This is capitalism.
[1]: https://www.youtube.com/watch?v=41VJudBdYXY