So why is the conversation centred around "rent"? Aren't rental prices just a byproduct of purchasing price?

You want cheaper rent? Make purchasing cheaper. You want cheaper property? 'Fix' the supply/demand problem.

I'm not entirely convinced that there is a problem - SF, NY and Sydney (heh, little old Australia still trying to stay relevant here) are great cities which are in high demand (close to infinite!), yet is extremely limited in space/land. To me, it makes sense that the price reflects that demand.

The talk is about rent because poor people rent and there are policy options that effect rental prices directly, rather than as a byproduct. Eg in NYC there are many co-op buildings that forbid renting units, or the idea of rent control, or housing vouchers or public housing.

I think that policy options that effect supply are the ones I would prefer, but we can't ignore other options.

The word "rent" covers more than one concept. The version you seem to be at is called "rack rent" - the price of a place to sleep. "Land rents" are a much more esoteric conceptual ... mess. You have to look to David Ricardo and Henry George for that.

SF is mostly 2 story bungalows and there are actually laws against building in the majority of the city.