If interest rates are up then you want a comparably higher yield (ie rent as % of purchase price) on the property (otherwise you could just earn interest on the money directly) and so that’s either a lower sale price or a higher rent. The rents you take in will mostly be determined by the market and it likely wouldn’t be a good investment to buy a house to rent it if market rents suggest you’d make a lower yield than buying similarly risky bonds.

It’s a bit of a different story if you’re just buying the property on the expectation that its value will go up better than other assets. There are reasons to expect that might happen (old people typically own more property and vote more than young people and they tend to care a lot about the value of their property going down so governments often have policies to try to ‘help people buy homes’ which allow people to afford higher prices which helps to keep the prices high) but it does seem that something has to give eventually (and eg it seems like there has been a bit of a change in Californian housing policy recently).

Generally I would expect an institutional investor to be more sensitive to things like interest rates and price when making purchasing decisions than a potential owner-occupier who has to live somewhere and may be more sentimental about certain things investors don’t care about.

It massively encourages investment firms to buy up housing stock from people whose mortgage is underwater and are forced to sell.

Investment firms keep liquid cash precisely for this reason, just in case a fire sale pops up and they can move quickly.

with wealth inequality what it is this seems more and more likely

If you have cash, buy when prices crash and hold.