Proof-of-Donation a compromise between Proof-of-Work and Micro-payments

'How, exactly, do you expect the browser to force me to donate to a set of charities, given that there are multiple open source options where someone can remove the "forced donation" code and re-release the browser without it?'

You absolutely could do that, assuming the browser even tried to force you. Then the web would fallback to work the way it does now, with many important websites regularly freezing up for a few seconds while forcing your browser to do Proof-of-Work, (or other nonsense like forcing you to do a Captcha just to view a single public facing webpage). Websites could instead deny you service entirely, but I imagine that would be rare. The webmaster might find Proof-of-Work wasteful, but if their goal is to rate-limit requests it may not make much difference to them whether they are limited by Proof-of-Donation, Proof-of-Work or Captcha.