They could try to offer some kind of "pay us to forgo marketing partnerships" deal, but I somehow doubt even most of the well-paid privacy advocates on HN would sign up.
I would pay for this, provided it was a one time payment! (I feel I need to add that caveat nowadays, since subscriptions are all the rage.)
The problem is that web browsers require some of the most intensive software maintenance of any software, ever. They're one of the most exposed attack surfaces to external security threats; and piles of new features are unfortunately getting continuously added to web standards, which need to be supported for compatibility reasons.
I think a one time tip-jar payment makes sense for a lot of software, but not for web browsers. I think a subscription-type contribution makes the most sense for them.
Subscriptions are just difficult to manage, I can't deal with individual recurring costs for so many pieces of software.
Wikipedia has sufficient marketshare to survive despite that.
The ads that mozilla shows are fairly privacy-friendly, much more than any other ads you'd find on most websites. That's an improvement.
If you don't like them you can easily disable them.
Users may not like nag screens, but I doubt they like the shitty ads Mozilla shows on the new tab page by default either. As long as the screen is easy to dismiss I don't see the problem.
Wikipedia has, as mentioned, the market monopoly advantage so they can do as please, Firefox doesn't and can't afford to annoy users.
I would doubt users are more annoyed by ads, especially the type of ads that Mozilla uses, than nag screens.