Most training organisations (especially non for profit aero clubs) barely have enough spare funds to paint the walls let alone fund a legal battle with young Johnny who accidentally hit the light pole on taxi.
For some it’s likely a case of simply writing off the excess bill and moving on, or going under.
The larger sausage factories with flashy pricey equipment might have legal representation internally, so expect someone to come after you. Those cadet courses on offer at the bigger players would have some long winded clauses in those contracts, read carefully. I doubt anyone signs anything at your local Aero Club.