> ...heads determining interpretations of law contrary to what the head of the executive desires.
Is the head of the executive an expert in all things? And capable of communicating those expert desires with perfect clarity?
Why have courts if the executive head can sort out all legal nuance themselves?
The Constitution is explicit that all executive power is vested in the president.
Article III courts can “sort out all legal nuance”, but the power remains with the president.
> Is the head of the executive an expert in all things? And capable of communicating those expert desires with perfect clarity?
It's not their job to be the expert, it's their job to be the decisionmaker. That's why you have a head. If two federal agencies want to interpret the law differently, it's more important to pick one interpretation and apply it consistently than to get it perfectly right.
> Why have courts if the executive head can sort out all legal nuance themselves?
Checks and balances are important but so is the ability to actually do things occasionally. Independence for the court system is good. Independence for every individual federal agency isn't.