Housing prices are kept in check when it is expensive to buy housing? This isn't necessarily false, but it is an obvious contradiction that highlights how complicated these issues end up being in reality.
No these two are not direct financial substitutions. You are ignoring a down payment (or mortgage insurance), property taxes, maintenance, and the general risk that comes with home ownership. Monthly mortgage payments should be a decent amount cheaper then rent payments on an equivalent property.
The other thing that's notable when you model the scenarios is how much of a difference it makes how long you stay in a house given the absurdly high transaction costs of selling in a house in the US. 5% realtor fees! $50k on an SF condo, $75-100k on an SF house.
Increasing interest rates erodes buyers ability to borrow, which will require prices to fall.