In general sure, but I don’t think so in this particular case. I really don’t think this one was that planned out considering they also sent these to judges who aren’t part of the executive branch at all (probably just because they have a .gov email). Many people in Trump’s admin have been reportedly finding out critical info via tweet, so the level of coordination here is often not high. In the actual email, there is no remark about “not firing equals resigning” (that’s only what musk said on twitter) and their legal filings for privacy & security stuff claim that responding to all the mass opm emails is optional.

Given Musk’s ego, I doubt he’d design something that makes him look weak like this.