But the other side of "Work just isn't the place for it" is this: sometimes personal problems affect you so profoundly, that they then affect your work performance. Your colleagues (and management) deserve to know about those things, so they can plan and adjust accordingly (and offer any personal or workplace support that is appropriate).
To use a somewhat silly analogy - we understand that if someone is too sick to work, then they shouldn't come to work. We don't tell people "work just isn't the place for your illness," because we know that some illnesses really make it impossible to do any work. And, that if you don't go home and rest, the ultimate impact to the business will be worse in the long run because you can't recover.
I think some personal problems rise to that level. Getting cut off in traffic and being annoyed? I don't know, maybe not. "The cops just shot 3 more black people like me at traffic stops this weekend and I'm having a really hard time concentrating now" might be a "personal problem" that is highly relevant to bring up at work (and coded as "political," to boot). That kind of thing would understandably rattle you, and it takes time to heal from.
I guess my point is: we're not robots, and it's not possible to really "check it at the door" most of the time. It never really has been.
The Basecamp policy isn't going to stop people from bringing their personal problems to work, overall. But it'll silence black people and gay people from bringing their problems to work, because those are "political" problems to some people.