But bond yields are based on a market.

If interest rates go up, bonds get sold (for better yield bearing products), pushing the yields of those bonds higher. And it finds some equilibrium. The fact it isn't immediate has to do with short term vs long term bonds. When they mature and the pace of arbitrage.

I don't see how a rate hike is meant to lower mortgage rate. And just looking at the figures shows it's the opposite effect.

Logically, if borrowing money becomes more expensive, how could borrowing specifically for the purpose of buying houses become cheaper.

You have to look at the current context. Bond yields have been spiking, mostly because of the inflation expectations from oil prices and tariffs (mostly oil prices). Mortgages mostly track 10 year yields, which is why when fed dropped the rates back to back, the mortgage rates didn’t come down. The current hike (and the next one) is supposed to create a deflationary pressure, but also provide confidence to the market that the fed will step in to cool inflation if necessary. This in turn lowers the yield on 10 year treasuries and therefore mortgage rates.

The fed rate provides a floor for mortgage rates, but the 10 year yield and mortgage demand decide the ceiling. Currently the demand is pretty low, and therefore the yield mostly controls the mortgage rates.