Then I think their problem is not “politics” but being unable to deal with individual employees who are being unreasonable and uncivil.

That's easy to say until you see how it actually happens. Basecamp would likely still be in the news, just instead of being about banning politics talks, it would be because they're somehow preventing their employees from supporting some cause or another.

This move is generating even more unrest and publicity for them

Oh for sure. Writing the blog post (at all) and putting that policy in without discussing it within the company was a REAAAAAAALLY bad/tone deaf move. When they announced it originally I assumed it had been implemented for weeks and they had the expected all hands meeting internally and concerns/edge cases worked out.

They clearly didn't. Saying that was dumb would be the understatement of the week.

It would have been instantly leaked by employees, might as well get ahead of things.

It would have, but after the employees had time to raise their concerns internally. Instead many of them felt kind of betrayed, as they learnt the new policy at the same time the rest of us did. Not really cool.