There are certainly skills involved in being a career politician, but these aren’t necessarily serving the public interest. I often feel like a politician’s main job is convincing constituents that their preferred course of action is best, rather than simply representing constituents interests.

Man, I wish that were the case. Convincing other REPRESENTATIVES is the main job of a legislator. The reason why lawyers are so good at the role of legislating (the nuclear engineer equivalent in your analogy) is that they are both trained in 1. Convincing others of their argument 2. Understanding legal standings and the workings of government. These skills should be the basis for someones eligibility to be elected. The reason we select one candidate over another is the ideas and values they represent for us in the day in, day out melodrama of governing. The only reason you think the important part is convincing constituents is because that is the part you see. The real work is making the damn sausage.