There are so many misconceptions in this post...
> why not put the pressure on the companies that are driving the influx of workers and out of control cost of living?
What do you hope to accomplish with this "pressure"? That these companies hire fewer people?
> Lower income housing?
Housing costs are one of those nasty problems where throwing money at it just doesn't work (and I'm saying this as a democratic socialist whose proposed solution to most problems is "tax the rich in order to throw money at it"). Imagine if big tech companies pledged a billion dollars or whatever to build a bunch of housing units. What would happen? This: other developers would pull back on new construction, because the pledge would throw a wrench in the supply/demand, and you'd be right back where you started.
The correct solution is upzoning and, to be blunt, disenfranchisement of NIMBYs.
> If tech companies are drawing people into cities and forcing out those who keep the city itself operating, why not have them subsidize and improve public transportation?
Most of the big ones do: they offer a lot of incentives to not drive, including paying for employees to take public transport or offering shuttle buses (which at least keep them out of cars).
> move their headquarters out of the city centers?
They aren't in city centers: 4 of the Big 5 (Apple, Google, Microsoft and Facebook) are all headquartered in suburbs. Only Amazon is headquartered in a city center.
> It seems crazy to me that people get driven out of their homes by real estate developers who re-develop due to tech-wages.
Why does that seem crazy? It's just supply-and-demand.