Martin, I'd suggest starting a new thread with a title like "What smaller towns can we see in the Dordogne without a car?" so you get input from more folks. Folks who've rented cars will not necessarily be sure about whether their favorite spots are accessible without a vehicle, but you can do an initial check on Rome2Rio to see whether there seems to be bus service from a nearby larger place. Just don't trust any of the details provided by Rome2Rio--not the bus fare, the travel time or the frequency. Use Rome2Rio as a starting point; drill down until you find the name of the bus company (supposedly) offering service. Usually there will be a link to its website.
I took a bus from Perigueux (handsome large town) to Brantome. Brantome was an attractive little place, but it wasn't really worth the effort. I arrived around noon and the weekly market had already closed up. I guess everyone was at home preparing lunch, because I was about the only person walking around. Not at all a bad place, but I think there are better options. Brantome is on flat land along a little river; it's not a hill town.
Sarlat would be a good base for a few days, I think. Even its Wednesday market was huge on a horribly rainy day; the Saturday market is said to be bigger. I would have liked to have a full day in town, but I balked at the cost of lodging and day-tripped from Bergerac (another attractive large town) instead. Take a look at a map of Sarlat to check the location of the train station vs. potential lodgings vs. the historic center. I recall a fairly long walk back to the train station.
Beynac is close to Sarlat, so I think it might be manageable by taxi. The distance seems to be about 7.5 miles. The trains weren't running at the time of my 2019 trip, so I took a replacement bus from Bergerac. As I recall, it passed very close to Beynac, but I don't think it went into the town. The TERs are running again, but there is no stop in Beynac. The question is: With rail service restored, are there still any buses into Sarlat that pass near Beynac? If there's still some bus service, It's possible you could hop off the Bergerac-bound bus and walk into Beynac.
I took a bus from Cahors (large town with nice historic center) that went fairly near St-Cirq-Lapopie. I was still left with a considerable walk, the last bit of which was uphill, but it felt like a safe walk. St-Cirq is very cute and atmospheric. I highly recommend it if the walk isn't too much for you and your wife.
I took a bus from Brive-la-Gaillarde (yet another nice large town) as close as I could get to Rocamadour. I had a very long walk (maybe 2 miles or more?), which turned out to be along a rural road with very little shoulder. Unfortunately, there was a beer festival on the outskirts of Rocamadour that day, so there was a lot of traffic on that road, and I was forever having to step into the weeds to avoid the risk of being hit. I would not recommend that anyone try Rocamadour by bus, because that walk was not safe. I believe the upper part of town (religious buildings) must be quite nice, but the lower part of the historic town is packed with cheap-religious-souvenir shops and tourist cafes. And tourists. It was a huge disappointment that my traffic-dodging had slowed me down so much that I could only see that unattractive area.
Also from Brive I day-tripped to Martel, an attractive small town that's definitely not as distinctive as St-Cirq. Martel is basically flat, so it might not be what you're picturing as a Dordogne village.
I suggest poking around on the internet to look for one-day bus tours going to multiple small villages like La Roque Gageac. I bet you'll find at least one option, but I don't know what the origin point(s) will be--maybe even Sarlat. Sarlat is very, very picturesque (and I'd like to stay there sometime), but it is very touristy, so there may be bus tours originating there.