Of course you can carve on a curved surface! It may be more challenging but not unreasonably so.
The reason to flatten is to have a known height/thickness and (generally) squareness. Flat and square stock makes zeroing and work holding straighr forward.
Assuming you want a curved final piece, you can use curved slats with some additional planning. If you model the curve, you can do surfacing pass(es) to remove the uneven parts enough to bring the rough barrel slats down to the exact model surface and then do whatever you’d like to the curved face.
It makes modeling and setup more complicated as well as adding some difficulty to work holding, yet still do able. You may need to make a jig/fixture to hold the stock to account to the non-flat bottom or curved sides. Align the slat + jig/fixture to a square reference. Zero off the known, fixed reference and model with the zero at that point instead of the stock.
This assumes your CAD/CAM software can handle non-square stock.