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.
the legislative branch can form administrative departments and prescribe their function however the president has already defined powers to impound funds and remove senior administrative officers and appoint/remove low-level staff. how these things intersect will be sorted be the courts.
executive actions (by-passing what should be legislation) have been increasing the last few decades. the various media companies plainly do make choices to portray some actions as nothingburger or crisis depending on their political alignment with the party in power.
the issue with the left-media and Trump is they outrage clickbait a bunch of events that are insignificant in terms of outcomes. Should they alarm about Jan6 yes. should they alarm over minor personnel at treasury or some dumb unserious thing Trump said at a press conference, no. This is how the media loses all trust in themselves broadly.