That is not going to work like they think it will, unless all they expect is chaos.
Well, to be clear, weekly reports are fairly standard in the private sector. What did you get done last week? What obstacles did you face? What's your plan for next week? Properly tailored to how people work, that's just responsible project management. On the surface this seems like a reasonable request.
You outlined most of the reasons why this particular stunt is stupid and wrong. Accountability is a two-way street. People need appropriate notice of how their work will be tracked and need to be provided methods that are compatible with their duties. The government is not just a big private company where everyone is expected to read their emails several times a day and also while at home off the clock.
We already covered why failure to do some specific thing cannot count as a "resignation." You can put it down as insubordination or dereliction, but if you're going to terminate someone for not reporting as instructed, that's your decision. Musk's email didn't include the warning, so perhaps someone who actually knows the law said something to him. This is not a genuine desire to understand government processes. This is just a feeble attempt at a flex. If some low-level guy suddenly has to justify his existence to the CEO directly, "or else," it's just meant to scare people who have otherwise been doing just fine.
There exists a chain of command precisely for the reason that information is best consumed at the appropriate level. In my business I don't give a flying flarp what some assembly technician did last week. I expect them to report to the shop supervisor, who reports to the head of manufacturing, and so forth. They have primary authority. My individual principal engineers report to me, but I work with them directly so I know what they're doing. Elon Musk likes to micromanage in order to feed the illusion that he's an expert at everything. If what you do doesn't make sense
to him, then he sees no use for you. Twitter showed us how that strategy wrecks a company, and now we why Musk's companies insulate themselves from his meddling.
And yes, obligations of secrecy apply both in an out of government. Classified work takes place over classified networks. Progress reports are classified. Since Musk was under investigation for violating his security clearance, he can't really be considered knowledgable about how they work and how to organize labor that must respect classification in instruction and reporting.
Elon Musk's official status in the government is still very much a matter of great mystery. He tells some people that he's the Grand Exalted Poo-Bah of all things governmental. But the administration tells other people that Musk is just an advisor with no authority. Heads of departments and agencies are quite within their authority to tell people not to respond to managerial requests outside of their chain of command.
But this is what they want. They want to create chaos and confusion so that any response to it can be styled as malicious. They want to issue illegal orders so that any resistance can be styled as the Deep State's resistance to the Democratically Elected President and contrary to His Majesty's Policy Objectives. Authoritarianism has been around long enough for people who want to do it to realize that mid- and low-level bureaucracy will try to preserve some function of government because they have more fidelity to their niche operation than they do to some distant overlord. Most people operate in government because they believe in the mission of whatever part of it they belong to. People like Musk and Donald Trump don't understand or believe this. To them, government means only power and kleptocracy.