> If a company has to sponsor you for an H1B though you'll be locked to one employer, and that lack of options is what means they don't need to give you market rates.

You're not locked into one employer on an H1B. Once you are here it is possible to switch jobs relatively easily since you do not need to go through the lottery again.

> as far as I know employment law leaves them quite free to offer a lower salary

"The H-1B employer must pay its H-1B worker(s) at least the “required” wage which is the higher of the prevailing wage or the employer’s actual wage (in-house wage) for similarly employed workers."

https://www.dol.gov/agencies/whd/fact-sheets/62g-h1b-require...

Ironically the Trump administration has made it harder to switch with the uncertainty they’ve been creating within the program.