Yes. But for the most part prices are relative. Your house increases. So does the move to house.
Higher price...insurace is more costly. Property tax...more costly. More capital gains (from a sale)...more CG taxes to pay.
You're generally correct. What you didn't factor in is all these things are relative. About the only way to win is to buy when mortgage rates are high (and closing prices lower) and then refi later. This will more or less cheat the relativeness of home prices.
Using cheap money to drive up price is a false god. It's a pyramid scheme. It's a - pardon the pun - house of cards. And now we're about to learn this. Sadly, again.