This is absolutely true in many cases. If I had myself to pay £2000 for journal submission, no way. But it comes out of research funds. And that in turn comes from whoever funds the University – government, or student fees.
Universities often limit what you can spend on OA, and even if there isn’t a hard limit, that money comes at the expense of other research items like student salaries and equipment. Funding agencies don’t write a blank check for publication costs.