You are right to question this trainer. Do not hire anyone who is not willing to explain her/his methods in detail. A good trainer should be willing to do demonstrations and allow you to sit in on a class (assuming that most trainers have classes in addition to doing one-on-one training).
Training is possible without food rewards. One of the exhibitors here who also competes in agility advised me to designate a special toy, one that you reserve only for training sessions. That works if your dog is toy motivated.
I have encountered too many trainers and dog owners who advise me to do things that will cause my Yorkies physical harm, including using a variety of collars.
Keep looking for a better trainer. I would insist on being present during training sessions.