Congress writes laws so it makes sense many politicians in that body are lawyers for the reasons you state. The court is made up not merely of lawyers but judges, their skill set is interpreting law, and dealing with their inevitable conflicts including legal hierarchies.

I don't agree the president needs to be a lawyer. That person can surround themselves with many good lawyers. The office of the president has the White House counsel, to deal with executive branch legal issues. The attorney general is completely separate, who is the head law enforcement official representing the entire government.

Ostensibly any issue arriving in front of the president is something important and difficult to decide on or someone else would have already done that. Good presidents have such decision making ability without getting stuck in minutia and petty arguments that serve no purpose, rather than acting like it's a TV show. They should have a firm grasp of ethics because often they're going to have to choose one among many good choices, or one among many bad choices. They should have a very good executive reasoning skill, cause and effect, consequences for their actions a decade or more down the road, rather than merely the next election, or the next Tweet. A good president needs to be a decent orator because they get their political capital from the people, and there will need to be many inspiring speeches, and condolence speeches. In this I'd include when to speak, and when not to speak.

Blah - there's more to it than that. But the point is, the president does not need to be a lawyer.