This would be similar to the central area of a mall or when companies provided housing to workers to prevent unionization efforts. Both places were upheld as places where the First Amendment must be upheld, despite private ownership.

A lot of people look at this issue and see two groups fighting over it. There are (at least) three groups though: for, against, and undecided. This is really a fight about how to convince the undecided, so solutions like "go make your own platform" aren't desirable; undecideds aren't going to know about the new platform, and won't automatically appear en masse on it, so it actually makes reaching them more difficult.