I pay $4200 for Los Gatos, CA (think more expensive than San Jose).

I have a 2 car garage and 2 bathrooms on a quiet street in a great neighborhood.

I'd randomly purchase a house in Pittsburgh before I'd pay $4500 a month.

Heck for 12 months of $4500 = $54,000 you could purchase a beater house in a worse neighborhood while you're trying to find your actual house, and you can burn it to the ground when you're done and still come out ahead.

Buy something for $200k that you know or strongly suspect is a stopgap and then start looking at your own pace.