Twitter is _the_ place where political discourse happens not just _a_ place where it happens. It's an unfortunate state of affairs that they happen to be a private company with a bias.
That still doesn't make it _the_ place for political discourse. Reddit gets quoted quite often. Tons of politicians do AMA on Reddit and Facebook. Just because Trump is on Twitter doesn't make it special.
That's not how these kinds of "facts" work. If someone tried to use this during some kind of proceeding the court will determine this. (Using a jury or not, expert witnesses or not, doesn't matter.)
Usually fact checking "services" like snopes.com, politifact, or WaPo with the pinocchio heads, have a consistent model about what they are willing to touch, how they approach it, and how they determine factualness, etc.
Just throwing out that Twitter is or isn't _the_ platform makes no sense.