The price of homes skyrockets. The interest on loans craters. The monthly payments stay the same. This is because the demand curve on buying houses is sensitive to the monthly payment, not the total price. So it all evens out...

... until you need the down payment. Then the higher prices/lower interest rates combo locks people out. It especially locks out first-time buyers.

You don't become homeless when you can't afford to buy a house. You become homeless when you can't afford rent (or your regular fixed-rate mortgage payment).

Rent is largely divorced [1] from house prices, and is largely driven by the amount of money people have after they get their paycheck, and subtract bills, gas and groceries from it. In a tight market, your landlord will take everything that's left from you. In a loose market, your landlord will only be able to charge you maintenance + his mortgage interest payments.

The best way to create a lot of homeless people really quickly is to have renters stop going to work, causing them to miss rent payments. The various stimulus and unemployment benefits cause some amount of main street inflation (increases in the cost of a basket of goods), but have also kept people in their apartments.

[1] Home prices in many metro areas are skyrocketing at the moment, but rents are falling.