I guess I'd be asking why is your DH that uncomfortable with sitters? And I'd side with you that his requests are a bit much. Background check, references, trial babysitting, all that I agree with and have done. I always use college age sitters just because that is what is here in my area, college town. I am FB friends with one on my former sitters (I don't use her currently because of her own limited schedule but nothing bad happened) and that is because she befriended me. I would never ask our current sitter to befriend me, I feel that is a invasion of her private life.
If I thought something was going on then I could set up a nanny cam, I actually do own some because of my DH's work but I have never felt the need to. I've gone through quite a few paid sitters over the years because they graduate, move on, etc... only one was I not sure about the care my children were receiving, in that incident I knew the girl's mother well, knew she had numerous younger siblings, so I knew she wasn't harming my children but maybe just not watching them as closely as I wanted. DS1 was 12 months old then had he fell off the couch while in her care, my thought was why was he on the couch alone in the first place. I simply never used her again and found another sitter. Now because we have almost full time care in our home, it would be tricker, but for a once in a while babysitters, it is easy to get rid of them.
I like for the kids and sitter to hang out in the home while I cam around so I can get a feel for how the sitter handles things. And I'm not afraid to ask my kids if they like them, what went on, I ask the sitter about the day/afternoon and get info. I live in a smaller town and I know a lot of moms! My sitters takes my kids out and I get feedback form others on how the sitter did.