I thought the SNCF *was* all the regional authorities and also the owners of the rail lines.
The SNCF is currently the only operator of regional train lines in France, but it's the regions that decide what lines are run and on what budget. They also own or lease the trains. The rails are indeed owned by SNCF Réseau.
There is also "Chemins de fer de Provence"[0] separate from the SNCF. I took it once in the 80s when I was kid traveling from Nice to somewhere in the Maritime Alps for a day. I remember a rickety old train that did not travel fast, but nevertheless had its charms.