For any postdoc that thinks the university has leverage for this reason, if you are able to get a H1B, you can switch employers easily. You are not stuck at whichever company sponsored you.
* My understanding is that universities prefer J-1 visas in almost all cases [1][2], and such visas are not portable.
* H-1B is more flexible than J-1, but still not as flexible compared to a green card. In particular, the new employer has to fill out a new application, pay the fees, etc. Also, universities are exempt from the H-1B cap, but other employers may not be.
[1] https://internationaloffice.berkeley.edu/ucb_departments/h-1...
[2] https://postdocs.stanford.edu/postdoc-admins/how-quick-links...
We also had an untenured research scientist who did all of the PI's work and grant writing. After two 110+ hour weeks getting three grants written for several million dollars they went home to enjoy the weekend and dropped dead of a heart attack at 42. I miss him.