Yes - agree here. When I’ve part leased my personal show horses out to riders that are more novice, I have always communicated (both in paperwork and verbally) that horse will get schooling rides (from me or pro) to maintain his schooling. I have learned the hard way to lose months of great top line and muscle letting those good tune up rides slip on a crossrail riders lease and won’t do it again. You get back a horse that you have to spend weeks, if not months, legging back up. OP has not clarified his/her level of riding and proficiency so, this could be the case (or not) - but regardless I do think it’s the owners right to maintain rides on the horse to ensure the quality stays there.
HOWEVER: I agree this is a case of poor communication. If the expectations aren’t meeting reality, OP should voice concerns with trainer and get it in writing (perhaps confirming via email or text in FU to a conversation) so everyone feels more comfortable! Communication is key.