Ask HN: How much power does the US President have?
Don't the un-elected employees have more power collectively?
Seriously, what do you mean by "power," and how would you measure that? The word encompasses too many things. The US President commands the most powerful military in the world, so in that sense he (so far always he) has a tremendous amount of power.
On the other hand the President cannot drive themselves anywhere in their own car, have a private phone call, or go anywhere in public without Secret Service agents supervising what they do. So in that sense the President has less "power" than I do.
The permanent bureaucracy you refer to has power over various things that the President may not have authority to control. I don't know how one could compare, for example, the power the IRS has to enforce the tax laws with the President's power to send a Seal Team on a secret mission. Asking the question is like asking if an apple tastes better than 50.
Are you familiar with the story of the bellwether? Collectives may be the body of power, yet it is a daring and vulnerable few who break rank.
There is the power of the role (powers explicitly described by law, which are too numerous to recount here) and those which come by the opportunity of influence (which are many more than can even be imagined.)
Even power of the head of state is fraught with trepidation. While this person could take sweeping actions, the covert backlash of resistance would undermine all future initiatives. The greatest costs of power are not those anticipated, they are those residual which fester covertly in the minds of those affronted.
Power is a fickle thing, and laws but give it consistency.