The logic is absurd.

"I should be part-owner!" oh, so when the roof needs to be re-done, are you going to pay for that? If the yard needs to be dug up to replace the water mains, are you going to pay for that? If the basement leaks, are you going to pay for that?

Oh, you're not?

Then the thing you're paying for is free use of the space for a certain period of time. If it isn't your responsibility, then you don't own it.