> And, it does genuinely seem weird to have an executive branch where the head of that branch doesn’t actually control things.

It's not weird at all, and it's not true that the head of the executive doesn't have control over those agencies. The head of the executive names their leadership - that is a huge amount of control. And it is enough control - the government isn't some top-down system serving at the pleasure of the president. It is a system for implementing the rules set out by Congress and the courts (subject to the President's judicial review powers), that the president coordinates.

The very title of "president" was chosen by the founding fathers to evoke the largely bureaucratic role they had in mind. It's not supposed to be a position of prestige or control like a dictator or ruler, it's similar to the role of a committee president: someone who oversees the functioning of the committee, and steer the general agenda, but who doesn't otherwise get to decide for the committee.