Let's start first by pointing out that there is (currently) no true answer to this question. Most of the powers of the Executive, Legislative, and Judicial branches are established by convention and practice, and are not explicitly spelled out in constitutional or statutory law. When the framers created the constitution, they intentionally left it loose and general: a literal framework, that future generations would have to fill out. And for the most part that 'filling out' has been done through practical, mutual agreement between the branches.
It has traditionally been the convention that presidents do not micro-manage the various departments of the executive branch, mainly because presidents are elected officials: non-expert politicians whose primary duty (assumedly) is to set policy. Department heads, by contrast, are (assumedly) specialists and professionals who are far more qualified to run the day-to-day business within their departments than any outsider. Presidents set agendas and directions, department heads create the implementations for such within the constraints of their professional knowledge and departmental mission, and then staffers carry out those implementations. That's a sensible system, if a bit bumpy at times. Non-elected careerists often have a different conception of their department's mission than elected officials, and department heads — who are generally appointed by an executive — often have to walk a cautious political line to both implement the president's goals and keep their department in a healthy state. But it's a system that has worked reasonably well in most cases.
This type of micro-managing also flies against the philosophical roots of our system, which has always had an explicit imperative against the accumulation of power in any single office. The founders designed our system to spread power over many, many offices and many, many people because they were concerned about the rise of tyranny. This is particularly true of judicial, military, and police powers, which the founders tried to distribute as widely as possible. A president who gathers the powers of investigation, judgement, and military force under his direct and personal control is no longer a president; he is a dictator.
Trump — though this hardly needs saying — believes that he has the right to do whatever it is he wants to do at any given moment in any given situation. This has little to do with him being president; he's a clinical narcissist, and narcissists feel privileged and entitled by nature. I do not know if he knows of or understands the established practices and conventions of our political system or the reasons behind them, but even if he does he surely does not believe they apply to him in any way, shape, or form. And because he has managed to push sycophants and attack dogs in as heads of many important departments (as well as grooming Mitch McConnell and Lindsey Graham as more-or-less mindless followers in the Senate), he may succeed in imposing his 'right' to micromanage and accumulate power on the nation, to everyone's (except his) detriment. If that precedent gets set, then yes, this will be a 'right' of the executive branch, and the US will transition fully into the status of a failed democracy.