I think your political views shouldn't, but producing bad pr for your company has clear, verifiable, and obvious damage to your company is grounds for dismissal. Many emplyment contracts already stipulate such things as that, and additionally, could be possibly construed as unauthorized communication in the name of the business. While I would not likely personally fire someone for posting such things, I would for the attitude that begat them. Self sabatouging people are a net burden on any enterprise.
I agree with your points. There is an intersection between free speech and contracts with a company --sometimes it's murky and it's best to avoid a conflict of interest --unless that's what you're looking for.
