My understanding is that Wikipedia does not survive on, or even need donations, at all: they are entirely funded through their endowment.
The donations go to the Wikimedia Foundation which spends the money on a bunch of other social stuff that's not related to running Wikipedia at all. So unless you want to support all those other causes, you're effectively wasting your money by donating to Wikipedia's frequent donation requests. Wikipedia isn't the storage hog that YouTube is; it doesn't cost that much to keep it afloat, and the people moderating the content (the editors) are all volunteers anyway.
While that is true and it is much more worthwhile to donate to the Internet Archive instead of Wikipedia (many Wikipedia references depend on being archived by IA), I would not want to start a anti-Wikipedia-donations movement. Sure they have plenty of money now and are wasting some of it and are begging for more, but if they would stop receiving any, they can eventually burn through everything they have. And having reserves and a steady inflow are important for planning future projects.
However, at this point, IA needs your donations much much much more than Wikipedia does.
From what I've read (I may be wrong), Wikipedia has enough in its endowment to survive indefinitely. Of course, if all donations suddenly stopped and they kept wasting money on non-Wikipedia stuff, this might no longer hold true, but that's an issue with bad management.
Bandwidth is expensive and videos take up way more bandwidth than text. I'd say it would be pretty much impossible to run something like YouTube, on that big of a scale, solely on donations.
Youtube sells ads and is subsidized by one of the biggest ad companies in the world that happens to have a lot of cheap cloud storage available.