Why not? Public institutions are supposed to serve the public’s interests.
Because I don’t want to give some unelected bureaucrats the ability to discommunicate someone because they said something stupid. Public goods are meant to serve the public, even if they have bad opinions.
I think the limit should be pretty high, but I’m fine with, as an example, people who spread abject hatred being rejected by most parts of society. I think not spreading hatred against your fellows is an integral part of the social contract.
What about someone who doesn’t think that transgender women are women? Should they be rejected by society for holding that view?