Yes, of course they did: design docs, RFCs, some PoCs to validate assumptions on small scale, etc. that's all part of a staff engineer job.

Now the question is: does Elon have enough technical expertise (or access to such) to be able to judge if a collection of assorted 10 screenshots without context of some GDoc pages, some shoddy PoC code and comments on RFCs is up to par with his bar of what a Staff Software Engineer title should produce? I highly doubt it.

It's not about showing evidence of work, it's that technical work can differ so much depending on context and constraints that it's impossible to demonstrate its value in such a reductive way, it's simple and clear stupidity to even ask to be demonstrated that way.