The NTSB (not the FAA) handles the part of aircraft accident investigations that you're likely interested in. They then make recommendations to the FAA on matters of aircrew or aircraft certification or other deficiencies that were in the causal chain of the accident.
They also don't travel to the vast majority of aircraft accidents, even ones with fatalities. They do travel to jet (or airline turboprop) accidents with fatalities, but small general aviation aircraft crash fairly regularly with only a small portion of those crashes getting any meaningful investigation from the NTSB.
I should have elaborated, but I sort of just tossed my post on here quickly. I was imagining if we get to the point that self driving cars become fully autonomous, and lets say 100% of cars on the road are self driving-- what happens if there's a crash? Theoretically that shouldn't happen right? If it did I would want an independent body to investigate.
They also don't travel to the vast majority of aircraft accidents, even ones with fatalities. They do travel to jet (or airline turboprop) accidents with fatalities, but small general aviation aircraft crash fairly regularly with only a small portion of those crashes getting any meaningful investigation from the NTSB.