If a shop or an instructor can't take 5 minutes to check out your gear and see that it's safe, they are not worth even considering. The insurance line is crap. They just don't want other students to see stuff they do not sell.
I never refused a student who wanted to use their own gear for open water once I had a chance to look it over and make sure it worked. If a student showed up with a double hose reg and horse collar BC I would have let them use it as long as there was a way to have an octo on it.
In my opinion, you having your own gear for open water is less wear and tear on mine or the shop's rental gear and it gives you a leg up on getting comfortable with the stuff you will actually be using.
There is little to no difference between PADI and SSI standards for what has to be covered.
However, PADI does allow, though doesn't really go overboard in supporting, independent instructors.
SSI does not allow independent instruction and their model is based on supporting the shop and selling you as much gear as possible.
The industry model is more and more geared towards supporting the business over the student in some ways. It's why I wrote my books and why there are separate entire chapters on choosing an instructor, shop, and courses.
Don't restrict yourself to PADI or SSI either. Look at the other agencies. SDI, NAUI, SEI, PDIC. As soon as someone tries to discourage you from using your own new, functioning, safe gear.. Run like hell.