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Yes. That's a key thing -- He's the executive, the principal executive. Every single employee under him is his agent, and is only acting under his authorization. This is where a lot of people mistake the tail as wagging the dog -- if they aren't doing what the president wants, then the employees are not completing their role and it only makes sense that they would be sacked. It isn't their job to second-guess him or work against him. (Now, that doesn't mean they can't or shouldn't provide counsel, but that's different than what we're seeing a lot)

Even that question of "lawful orders" doesn't mean individual employees get to just go "oh, we're not doing this because it's not legal" -- there's pathways where checks and balances see if you're right.