If everyone flees $5000 rent, wouldn't the rent then come down ?

In a normal market, yes it would.

But there is literally no residential construction in most parts of the Bay Area.

Nobody lives there anymore, it's too crowded.

Yeah, but they've been saying that about tech since the late 90s, and about California in general since at least the early 80s, where my history begins. Supposedly they've been saying it since the 50s or 60s.

Or Portland.