What kind of horse should have a full cheek snaffle bit?

Its is not the "kind" of horse that determines the type of bit you should use it is the discipline and training level of the horse that determines it.

A full cheek snaffle should be used on a horse being ridden english. It is a fairly mild bit that can be used on horses of all training levels. The cheeks pieces are there mainly to prevent the bit from slipping from side to side in the horses mouth but it does provide a bit more action in the horses mouth than a loose ring snaffle.