It’s not that complicated really. House prices tend towards the level where a mortgage payment roughly equals a rent payment (since they’re substitutable). So when mortgage rates go down house prices go up commensurately.
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.