This shouldn't be the case. Youtube has a monopoly on digital video as a media and nobody else even compares. We don't think of them as a brand--just a thing now.
YouTube is not actually a monopoly. There are no market barriers preventing someone from making a website that functions exactly like YouTube and provides the same services. In fact, there are many sites that do that.
What you're talking about is the fact that among all video hosts, YouTube has the largest
network effect. People use YouTube because
other people use YouTube, and the popularity makes for free advertising. It's like if you were a painter and wanted to show your work in the largest, fanciest gallery in New York City. If that gallery bans you for being a bigot, then you absolutely lose out on a large potential audience, but you aren't prevented from sharing your work with others. There are thousands of galleries that will still take you.
I also don't think Google should be allowed to get rid of search results about Annoying Orange because they supported Clinton, or allow ComCast to shut down pages related to free internet because they don't agree with that on an economic basis.
It's not whether they 'should' do that or not. They
can't. Like it or not, a stuffload of people supported Annoying Orange during the election, and Google knows that they would see backlash and boycotts on a massive scale if they were deliberately censoring out search results related to Annoying Orange.