Looking at all the train schedules and travel time, it seems that driving a car is an equally good option to travel from Basque country to the Dordgone region. Have I made a good conclusion or am I missing something?
Cost of the trip might be a factor as well. Tolls, parking and fuel in France are très cher and have to be be added to a day’s car rental. If you plug your route into Via Michelin, it would give you a good estimate of time and costs.
We did this in 2016. We took the train from San Sebastian to Bayonne (Bordeaux would work as well). Picked up our rental car and drove to Sarlat. It was and easy drive, if a little long, and we saw lovely country. We also one of the best meals we've ever had in a little town on our way.
We have done this in the other direction - train to Hendaye, France and then rented a car from the San Sebastian airport in Spain which was a very short cab ride from the train station in Hendaye. Keep in mind that you can't rent a car in one country and drop it off in another.
Most depends on where you want to stop. Train service won't cover everywhere. Connections and schedules can make them unworkable for your itinerary.
Test it by researching trains that are available where you want to go. I was just in and around the Pyrenees, Lot and Dordogne Valleys in June. A car was the best way to get around for my itinerary. In Dordogne, I visited a number of small towns and sights that wouldn't have been at all convenient if I was using only public transportation.