This is a nice hypothesis but fails to account for the reality on the ground - foremost being transportation.
So you are displaced out of SF - great, where to? You still need to be able to get to work in the city - after all, it'd be pointless to move somewhere cheaper just to end up out of work.
So, let's head south. Oops, due to the overflow of demand in SF, as well as the sheer number of tech workers looking to straddle the peninsula and the city, prices down there are only slightly less insane than in the city proper.
Okay, let's head north - well, nope, Marin is not exactly poor man's territory.
West is water. Humans float, but not that well.
Let's head east - Oakland is rapidly being gentrified and colonized by tech workers also. On a relatively low income you can probably still scrape by - but not for long, in a year or two you'll be priced out yet again.
Further out? How do you get to work? The toll on the Bay Bridge is ridiculous, and it's not like you can afford parking in the city so you can work anyways. Your only options are along the BART, but park and rides are few and far between, and properties within non-driving distance of BART are all rising rapidly in price. Not to mention, BART fares are distance-based (rather than flat on, say, MUNI) so the further away you move the more expensive your commute gets! For a tech person making six figures a year the price difference isn't even noticeable - but for someone scraping by?
This whole problem has played out this way because San Francisco (or really, the Bay Area in general) is the most frustratingly myopic place I've ever lived when it comes to transportation. Unlike cities like Chicago or New York (or even Seattle, which is primarily bus-based), the transportation network is in such poor shape that it actively penalizes the poor. The well-heeled, like ourselves, can afford cabs and Ubers to bridge the gaps where the transportation grid fails. The poor cannot.