Unfortunately the act of pulling the outlet out of the box caused some wires to disconnect, so I don't have any certain idea how they were wired initially. It's clear which wires connected to the scorched part of the outlet as they are scorched too, but beyond that, nothing is certain.
In addition, I have not been able to find anything else nearby that doesn't work. The outlet for the washing machine is the only other outlet in the laundry room, and it is still functional.
It sounds like you're saying there's no common reason to have additional wires coming into the dryer outlet. That is, I shouldn't have anything more than the strand of three heavy gauge wires. Is that right?
Here are some pictures, in case that helps.
The smaller wires are red, white, black, and ground (no insulation wrap - just straight copper). The larger wires appear to be aluminum - 2 are black and the 3rd has no insulation (ground wire).
I have an electrician coming later today to check things out as I'm feeling a bit risk averse and unsure about what is going on with the wiring here. I know it's generally a bad idea to connect copper and aluminum, but that is apparently how this was wired when I bought the place (this is not the only example of strange/amateur work in this place).
Is it safe to run a 220V 50Amp outlet on the smaller gauge wires? I would have expected such an outlet to require the heavier gauge (like my aluminum wiring).
For completeness, I'll document here the final resolution of this issue. An electrician came by and we identified a 220V outlet in the garage that was the reason for the smaller gauge wiring in the box (the red/white/black/ground strand). The to this outlet was done improperly and we have no need for this additional outlet, so I just decided to cut these wires off and I'll eventually remove the garage outlet altogether. With that out of the way, it was very easy to hookup a new outlet using the remaining, heavy gauge aluminum wires. Case closed.