They can choose to not be a platform for anything that they want. You don't have a right to that platform. Forcing YouTube to leave everything up infringes on their rights to run their site as they want.
Now you could argue that Donald Trump expertly exploited exceptions to the "equal-time rule" in order to get much more coverage than his competition, but that's more to do with the structure of the law than the spirit of what it's trying to accomplish.
Youtube, or any other site uses no public property in that way so they are not, and should not be beholden to the same rules.
YouTube is just asking to be regulated in such a way as the phone companies. And it should be regulated to follow the First Amendment as a mass communication platform, even though it is a private company.