Thank you for the replies.
JRC, it isn't the paint. For one thing, it's crooked, and we've been unable to get it aligned properly. More importantly it does not latch properly (barely at all). I only just realized that these are mortise locks and, as such, there is quite a lot going on inside the door that's hidden from the eye... (I just got back from a wonderful hardware shop where the owner showed me the inside of an old mortise that someone had brought in - pretty intricate!) So I will try your suggestion of opening that up and cleaning it out... There is a decent chance it hasn't been cleaned in at least 50 years. And/or, there could be pieces rusted and/or broken, as was the case in the one I saw at the hardware shop.
We actually tried repairing the front door knob (pics not shown), including replacing the spindle, with no luck. I think we will have to pull out the whole assembly and see if we can possibly fix it ourselves. I have looked but, so far, not found anyone local who works on antique lock sets.
I think we might have the wrong spindles and/or set screws and/or even knobs on some doors. Everyone we have talked to seems to think it's pretty simple business, but we've been unsuccessful so far. It is frustrating being unable to figure out why they don't work and/or the knobs fall off.
We have ONE door that works perfectly. Although it feels like tempting fate, maybe we'll take that one apart to see all the proper parts and how it works...
Lynners, thanks so much for the link. Wow, that one was dirty inside. I do wonder if many of mine look like that.
The guy at the hardware shop mentioned using "liquid graphite" for oiling the locks inside - is anyone familiar with that?