And that in large measure IS within the purview of the President. The 'federal bureaucracy' is the Executive Branch. If Trump wants to reduce the size, scope, cost, and corruption of government the Executive Branch is where he can start.
Again, all legislative power, according to Article I Section 1 is vested in the Congress.
Congress needs to, by law, take away the law-making power of the bureaucracy. That will, in effect, shrink it...
President has a role to play of course...in effect, enforcing the laws Congress has passed.
I think we need a re-alignment from statists (currently in power in both parties) to a more representative republic-—but we ALSO need a re-alignment of power back to the elected representatives of the people rather than unelected federal agents.
The IRS, the FBI, and the CIA create “fear” in people because they are often allowed to run amuck—without a check on their power. Time for the Congress (esp. the House)—directly elected by the people every two years—to re-grasp that power.