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.
To play devil's advocate to the "private platform" argument: we force tv networks and radio stations to give equal air time to all presidential candidates.[1] NBC is a private a company but if they permit one candidate to host SNL they must give the other candidate(s) an equivalent opportunity so as not to unfairly sway public opinion. I don't necessarily think that "infringes" on NBC's right to run their platform as they want. They're still given the option of whether or not they want to wade into politics on their airways but once they do they're obligated to open their platform to both sides.
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.
Broadcasters have purchased a license from the government to use those airwaves, and have to abide by rules. In contrast, anyone can put up a website. That is why broadcasters are more constrained in their behavior.
The issue with TV is that it's not "their airways". The "airways" are public property so to speak, and the TV license grants TV networks use of that public property on certain conditions, one being political messages.
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 has become a mass communication platform, more akin to the telephone company than say a magazine. A magazine has editorial control over what it publishes, and can decide what to put into its content. A telephone company has no say over what is said over its phone lines.
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.
