What makes the Dordogne so attractive is that it's extremely rural, with dozens of small villages, castles, and caves lost in the middle of nowhere, inaccessible except on foot, by bicycle, or by car.
There are no Uber drivers in the Dordogne; it wouldn't be useful; there wouldn't be enough people to transport. Locals don't use Uber, and many don't even know what it is.
There are very few taxis; these are mainly used for medical transportation, such as for elderly people in retirement homes who have medical appointments at a hospital or doctor's office (taxi fares are paid by the social security system). Taxi companies are also often private ambulance companies, as operating solely as taxis in the Dordogne is not profitable enough.
The bus network is not designed, with a few rare exceptions, for tourist routes but for locals, and especially for school transportation with the resulting routes and schedules.
So, to summarize: A tourist without a car has no chance of visiting the real Dordogne, even with a travel agency that will only show a glimpse of a few places for an exorbitant price.
As for Sarlat, it's one of the largest towns in the Dordogne, with less than 9,000 inhabitants. Over the years, it has transformed into a kind of visitor center overrun with tourists, many of whom don't go further than Beynac-et-Cazenac and La Roque-Gageac, which is about 10 km from Sarlat, with the exception of Lacaux IV, which is 25 km away. The Dordogne Valley is over 150 km long. Sarlat is located at the far west of this valley, but the landscapes, castles, and villages are identical all along the valley, as well as in the Lot Valley a little further south.
Any good travel guide will tell you about it.