To live in the city, you really need a household income of $300K+. Anyone below that is either going across the Bay Bridge or south into the peninsula.
Based on your pricing, you're best off heading to the East Bay and commuting in via BART. The peninsula may be a better option but you won't get much relief from the rents and everything is car-based and a bit sleepier for young people.
You are so right, we live in San Mateo now and the closest place we could afford to buy a house without maxing out our budget was Tracy, CA (Pleasanton/Livermore are is also still quite affordable). Needless to say the commute is terrible from there so we have rented the place out and hopefully we can return there once our jobs are more settled and we have more flexibility to work remotely.
In the limited research that I've done about Bay Area real estate, you're best off selling the Tracy home while prices are high, because the schools in Tracy aren't great.
There is likely a minor market correction coming but you could see a lot of the homes lose 20-40% of their value in places like Tracy. The homes that will retain most of their value will be the ones with really stellar schools (9+ on greatschools.org or higher) because there's always demand for them. They won't correct much beyond 10-15% at the most.
If you're dead set on buying, I would go with Pleasanton if you can afford it. Solid schools, and a somewhat reasonable BART commute into the city. Otherwise just save your cash until you can drop $1 mil into a house.
Based on your pricing, you're best off heading to the East Bay and commuting in via BART. The peninsula may be a better option but you won't get much relief from the rents and everything is car-based and a bit sleepier for young people.