>To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.

So yes, Congress has wide leeway to create agencies to administer particular executions of laws passed by Congress and signed by the President.

Furthermore, we don't have untouchable officials. What does that hypothetical have to do with anything?

The heads of these agencies that Trump is removing are protected from being fired immediately, fired without Congressional notification, and from being fired without cause.

Rank-and-file civil servants are protected from being fired unrelated to performance precisely so we don't revert to the patronage systems of the 1800s.