I don't think making homes more unattainable is the answer.
Yes, higher interest rates would make the buyer pool smaller, which in theory should lower prices, except inventory hasn't been this low in 17 years. That's the issue with making money more expensive is you make it more expensive to build and that drives the supply problem.
Also, higher selling prices help with selling your home but then you need to buy another home.
This doesn't get into what's happening in the rental market... Friends of mine are not able to afford their increasing rent (up 30%), they can't afford to buy in the neighborhood they currently rent in, so what now? Buy/rent in the suburbs. This is what we want?
Supply is THE issue, suppressing purchasing power with higher interest rates only seems to feed into the rental problem.