I have been working and hiring remotely for years, and that is generally false.
First, if I have a perk, I will list it. It is a competitive market for talent. This is just another tool at my disposal. Would you say that about "unlimited time off" or "generous benefits" being advertised in job posting? The same pool of money pays salaries as team outings and office beer taps, catered lunches, and other perks. On a basic level everything is zero sum.
Second, as for your "across the river" example, to the extent that actually happens, that view is a bit SV-centric. For talent that wants to avoid overpriced homes and soul destroying commutes, their frame of reference is local salaries that are guided by local costs of living. As a prospective remote, I don't care if you pay +20% in SFC. I am in another place, and if you can outbid my next best offer by 10% I win.
Definitely everyone should negotiate and be aware of salaries in NYC, SFC, Seattle, and DC.