Pilot DAR,
In the UK over the last few years there has been very noticeable increase in the likelyhood of a student/customer, their dependents or a even a third party starting litigation in the event of an incident.
As I said in an earlier post, my experience is that the aircraft/club insurance takes up the case on behalf of all the defendants.
The concern partly comes from the perhaps grey area, where the FI is a self-employed sub-contractor, rather than a direct employee. Also some flying schools have been known to be run by unscrupulous individuals, who may not actually have liability insurance in place and lie about it, if you have your own, you know you're covered. I once had the misfortune to work for a bit of a crook, when the company went into administration, we found out the insurance company was one of the creditors as they hadn't been paid the current year's premium. So if we'd had an incident there would have been no cover in place, as the certificates that were at the school weren't valid.
You also have the possibility that if the flight was illegal for any reason the club insurance could easily wash their hands of the PIC, although Besso's might well do that, don't know what their Ts & Cs are.
With regard to the post qualification scenario you mention, I am aware of a case where a PPL holder crashed and killed himself immediately after receiving his licence. The family sued the instructor for not training him appropriately IAW the syllabus (which did happen to be a correct assertation in this case). There is not a lot of direct oversight by the UK CAA of Registered Training Facilities to be honest. There is quite a heavy reliance on instructors and particularly examiners upholding the standards.