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 doesn't need to be a coup. Congress sold us out to presidents long before most of us were born.