Whelp, on the advice in this thread I attempted to rehab my old freewheel and it actually went swimmingly. The rust-coated gears make it look a lot worse than it actually was; once I used some WD40 to clean off the oily grime in the middle of the freewheel it was squeaky clean and in very good shape. Thirty seconds later it was spinning free and bathing in some 3-in-1 oil. The shifters were actually exceptionally clean once I took the covers off; after cleaning and lubing them they worked well.

With the wheel back on I gave everything a test run, and with the rehabbed shifters the derailleurs worked smooth and nearly flawlessly, with the exception of the rear derailleur sometimes missing the second to largest sprocket. A minor inconvenience, but the important part is that the drive-train is in working order even with no replacement parts. As for the brake/derailleur cables, they obviously aren't in brand new condition, but they aren't rusted either and they all appear to be working fine.

I'll still take it by a shop to have it checked out, but it looks like the only thing it absolutely requires to be ridden now is a seat and some tires/tubes.
Thanks everyone!