The wording seems to be about "mechanical failure" though. If your car catches fire on its own, whilst not being driven, then presumably that's probably down to electrical failure and not mechanical.
I guess the insurance companies are trying to make sure they don't get involved in anything mechanical, after all, they don't cover breakdowns.
I think it would be very difficult to get your own insurance company to have to pay out. I would have thought the best bet would be the seemingly rather large coincidence of the car just being driven away from the garage after having been worked on. I would have thought that a court would be very interested in this "coincidence". The problem is trying to prove this, since it may be expensive to get an independent report into what caused the problem.
Thinking out loud, I wonder if Kwik-fit could be persuaded into some sort of cash settlement where they don't actually admit liability, but which ends up being better than nothing
Might be worth getting Trading Standards involved, on the basis that you had work carried out and then immediately the car went wrong, so the standard of work doesn't seem like it was too good ? This might persuade Kwik-fit to start being a bit more cooperative.
I do hope that it works out well though, sounds like a tricky one.