Hello!
I don't know what this insurance thing is, that comes up with almost every thread. The only way an insurance can refuse payment is by
proving, that the instuctor or pilot in command has been
grossly negligent or even acted deliberately to cause damage.
No court in the world would rule for gross negligence, if a student - who for years is used to starting the engine of a car, motorbike, boat, bus, lorry! - and has started the engine of an aeroplane a dozen times under the supervision of an instructor, is allowed to do that on his own. It's really not rocket science. Otherwise no instructor in the world could ever dare to send a student solo!
That said, I still wouldn't do it. Simple because I am also the kind of PIC who insists on doing his own quick walkaround (too many oil dipsticks found untightened or overtightened...).
Happy landings
max