I'm lucky enough to have too many properties to hunt. I don't say that arrogantly, it's just how it worked out. I moved here from the city and now have a co worker, friend I met at a local airport, and a neighbor that have property. None are rich and only one likes to hunt.
I can see your side though, our family property, and even extended family property is only hunted by immediate family. It's nothing personal, we just have enough problems with poachers and trespassers as is.
I think that is your best angle - I was hunting Sunday morning and heard several shots that had to come from behind the house. Nobody else in our family was hunting, so it was probably poachers. Most landowners like to have someone they know around or in the woods.
I also 2nd the point about putting in time. One property I went up and cut hedge trees and honey locusts on a fence line all day. I only helped out this one day but it made a huge difference.
You may also offer to start out bow hunting. It's more difficult but that's how you can get someone to trust you and once they see you come and go a few times they might just let you bring a gun.