exactly, think of a bartender refusing to serve a problematic former client a drink, or the bouncers not letting them in, due to them being specifically sanctioned.
private business absolutely has the right to refuse service to people over their behavior or expressed intentions.
the US first amendment protects against GOVERNMENTAL infringement.
in terms of this Twitter tempest-in-a-teapot, they ALSO have a right to free speech and Trumps demonstrably FALSE claims can absolutely be addressed, labeled as false, and that is an absolute right to free speech that Trump has already threatened with specious "governmental action" which PRECISELY violates both the letter and the spirit of the first amendment!
Trump is violating it!