How familiar? Do they ever look at the .S files? know what ar and nm do? Deal with objdump? How important are those details for day-to-day compiling of C code for regular programmers?

Yes, they know what ar and nm do. Whether these things are relevant for day-to-day compiling of C code is watching the goalposts sail over the horizon; the original statement being mocked here is "we don't need to learn ASM because GCC is so good", which completely betrays ignorance of why people learn still learn and deal with ASM to this day. Even if LLMs were comparable to compilers either in predictability (they're not) or code quality (they're not), the people who know what they're doing still take pains to understand how their tech stacks work, because that makes them better programmers, and this will continue to be true.