After washing dishes in the kitchen sink last night using water hauled in big bowls from the tub spout -- two days worth, and I did a lot of cooking Sunday -- the leak is strongly correlated with using the tap in the kitchen, but not with running water down that drain and it doesn't appear to be dripping when the tap's not running.
This is relatively good news. The previous homeowner did his own plumbing, with a fair degree of competence but a complete unawareness that you can't intermix soldered copper and threaded galvanized-iron pipe, despite the existence of simple adaptors. Run water through it and there's some electrochemical activity between the two metals; the iron pipe rusts at an accelerated rate, the copper goes green early, and eventually it forms pinholes and weak spots. There's one of his mixed-metal improvisations under the sink, with a shutoff valve that feeds both the sink and (still defunct) dishwasher and the iron portion is rusting. (There's a right way to do this when you have to, a "dielectric union" with an insulating material between the copper and iron. But it's best avoided unless you have no other choice.)
Up top, I have a "coronavirus improvisation:" the actual spout of the faucet had developed small leaks early last summer and I gave it several wrappings of self-amalgamating tape rather than replacing it -- the shutoffs are iffy, one of them being part of the rusting assembly, and I wasn't confident I could identify the right flexible jumpers for the replacement fixture and I didn't feel like having a plumber in just then. But it leaks, pooling around the base of the faucet and running into the sink basins. I'm not confident it's not trickling down the plumbing connections inside the cabinet, every so slightly, and I'm just not finding it.
The solution to both of these is to replace the whole thing. Th pipes that come up under the sink are 1/2" copper. I can buy "SharkBite" push-on fittings (which I have used before) and replumb the shutoffs, distribution and end sections where the hoses for the replacement faucet screw on. I already have a replacement faucet and I have replaced them in other houses in the past, though it's been a dozen years or more. I will probably have to buy a small collection of parts and a new "basin wrench," a wonderful specialized tool that helps with the hardest part of the job.
I ordered a cheap video borescope that should arrive soon, so I can check in the hidden area between cabinet bottom and floor and see if there's anything lurking there without having to do a lot of drilling and sawing. And just in case, I ordered some inexpensive plastic washpans: if I'm going to have to use water from the bathtub spout to wash and rinse dishes anyway, I might as well sit on the floor, put washpans for wash/rinse/drain in the tub and do dishes there, and not be slopping gallons of water on the floor. Yes, it's awful (and I plan to be using paper plates and eating takeout dinners until I fix the kitchen sink), but it should do for awhile.