Have you heard of the courts?

The president can settle any issues between their view, the agency head’s view, and the agency’s guiding laws, in the courts.

The courts are their specifically to settle disagreements.

That’s how the law works for agencies. That’s how the law works for all laws.

Three branches checking each other.

They are accountable to the courts, whose job it is to decide if they are faithfully fulfilling the laws laid out by Congress.

Again, they're accountable to the laws that regulate them and the judicial system, in their every action.

It's something extremely common in normal democracies, and if it's really antithetical to US civics, that's just another proof that the US were only a pretence of a democratic system.

Congress