The wise instructor will insist that the policy reflect that instruction is being given in the aircraft, and that the policy cover the instructor. Maintenance is absolutely a concern, and a wise instructor will look closely at the maintenance currently on the aircraft, and it's history. Any agreement needs to ensure that the students will operate strictly in accordance with the instructor's direction. That means no freelancing, no taking the aircraft to go fly when you feel like it, or doing anything outside the authorizations and limitations set by the instructor.
Any of those represent dealbreakers to a wise instructor.
There is often a sense by an owner that because he is invested in the airplane, he will set the terms; make no bones: the instructor is in charge so long as he's providing the training and you're operating on his certificate (which you'll do until you reach certification. The instructor directs the operation, not the student.