I think that's a big mischaracterization of what happened.
For one, it seems a lot of the issue was about the 'funny names list' and heated debate around that. It wasn't people going to war over liberals vs conservatives. It seems DHH took particular umbrage at someone brining the Anti Defamation League's Pyramid of Hate into the conversation.
Secondly, Basecamp had allowed the creation of a Diversity and Inclusion committee with at least a dozen employees joining. DHH and Fried decided to unilaterally dissolve it. If you're going to give employees your blessing for a D&I group then just axe it with no discussion or warning, some people will be put off.
Then there's the fact that employees found out about this group of changes via blog post. That betrays a lack of empathy/care for employees when implementing a set of big changes.
Lastly, saying 'no politics or societal issues' because you, as the owners of the company felt uncomfortable, is a recipe for ruining the culture of a company like Basecamp. For some employees, they can't get away from politics and societal issues because it affects them every day and their very existence has been politicized. More savvy leaders could've established a better climate of respect and politeness around any 'political' discussion rather than a heavy handed and clumsy edict.
This wasn't about a bunch of people endlessly bickering about politics at work, it's about company owners who took a manageable issue and turned it into a public crisis. Companies several orders of magnitude larger manage to accommodate employees having political conversations without making messes like this.