The cost of the home is normally quoted in dollars today (what the seller receives), not the total dollars you will pay on your mortgage. So interest rates don't have much to do with affordability.
The typical early stage buyer is financing their home and very often buying within 25% of the mortgage they can qualify for, especially in competitive markets. Mortgage rates absolutely drive this affordability.
When you’re bidding against other buyers, the more they can borrow for a constant monthly payment, the higher the equilibrium price will be.