It's a bit of a tricky situation this, if a customer takes out the front end of one of your aircraft, that can as Whopity suggested wipe out your whole profit for that year. The margins in PPL training in the UK are pretty small, nobody runs a flying school to get rich.
I've (so far) never had a student do write-off level damage to an aeroplane, however we have had the usual strobes/wingtips etc, worst thing was a damaged nose oleo, which cost a few thousand to fix as I recall. We just bit the bullet and paid to have these fixed, if you run a flying school or rent aircraft for long enough, eventually someone is going to break your aeroplane, that's just a fact of being in the business.
There is no "special" training risk insurance that I'm aware of, you just go to a broker the same as any other owner and tell them you will be renting/training on it, all this does is change the premium and excess. I think it is a good idea to have a "prang fund" or similar, which everyone can contribute to, which can then be used to cover the costs of such an incident, without having to ask the individual for a significant amount, as they will probably feel bad enough already. However, insurance is not the complete answer, as by the time you deduct the excess, loss of no-claims and increased premiums in subsequent years, it's not worth claiming for less than about £7-8k.
As to who should be held responsible, well student pilot or not, when solo they are the commander of the aircraft, so isn't it down to them. How responsible can the instructor be, when not in the aircraft, as long as the student was competent, consistent and briefed and the weather was suitable. You just wouldn't ever send students solo, if you were responsible as the authorising instructor for anything and everything they may do. Ultimately, I think it just has to be the school that takes the hit on this kind of thing. As has been said, different altogether for PPL holders, they should be liable for any damage that occurs as a result of their action or inaction.