That's not how it works in the U.S. If an executive branch department was created by the legislature, it is up to the legislature whether or not it exists, not the executive. If the legislature has passed laws regarding how its resources are to be used, its employees treated, the executive is not free to disregard those laws.
The legislature is the source of laws in the U.S., not the executive. The irony is that the Republicans control the legislature as well. They could pass laws to achieve what Musk wants. It would be slow, but it would be legal.
A coup is seizing power outside the legal mechanism for doing so.
It's beyond me how so many of us think that continuously ignoring the will of the people is "OK". Either tell me my choice doesn't matter, or just shut up with the drama and enact safe and fair referendums on every single hot topic so we can all get to the right answer and then if we find we're in the minority, we'll shut up.
It should be clear as day to anyone that is unbiased that fixing the US/Mexican border was ridiculously easy (it's essentially been done in 2 weeks and they didn't even have to finish building their stupid wall). The only reason it didn't happen till now was precisely because the whole thing is broken and not really an expression of the peoples' will. It was rather an expression of an amalgamation of a giant mindless mass of bureaucrats, and you can't fix it unless you do what they are doing now. Not to single you out sorry, but opinions like yours ("we gotta do it the legal way and according to rules x, y, z, and 500 other rules") are precisely why nothing ever got done or fixed properly. And I say that as someone that is absolutely on board with following every rule to the T, with no exceptions.
I agree that our system of government makes it extremely difficult to enact large changes. That is by design, however well considered that design might be. Nevertheless, those are the rules. Which means the president can't legally do whatever he wishes to anything "under his purview" upon gaining power.
Or rather, that was the case until the SCOTUS decided there are no laws the president need respect. What they have not pronounced upon is whether the law binds anyone acting under the direction of the president. Does their invention merely protect the president from prosecution or does it abrogate all laws he finds inconvenient? I find it hard to believe they'll take the second step, but we'll probably find out pretty soon. Is Musk a monarch or merely our president?