Housing prices going down only matter if you want to sell.
Sure rents might be cheaper than owning, usually not though. Landloards are greedy as fuck. Buying my home was a good move because rents have gone up a lot since I bought it in my area. And of course rents could go down but... haha I joke.
Regarding mortgages going up after you buy: that's only going to happen if you have an adjustable. I have a standard fixed mortgage it will never go up.
Certainly my taxes may go up, but that only reflects rising home value and I'd get that out if I sell.
If we had reasonable renter's protections it would make less sense to buy, but outside of Mamdani's New York, renter protections are absolutely horrible in this country and buying is the only way to "opt out" of the exploitative landlord system.