I've also seen that programmers who're not really good at the job, but don't have qualms about playing politics often end up managing the teams they were unfit

I've also seen that programmers who're not really good at the job, but don't have qualms about playing politics often end up managing the teams they were unfit to work in in the first place. They were not necessarily better at people than others. The idea that programmers are not good with people is, in my observation, a self-perpetuating stereotype. Mostly because if you're writing code, you are on a maker's schedule and you don't have much time or bandwidth to think about anything else. Managers on the other hand have more control over their time and can for example spend an hour to decide exactly how a conversation should go before they speak to someone. They can reflect on it, and deliberately improve over time. The programmer on the other hand would be figuring out why a particular technical decision was wrong and how they can identify it next time etc.