Count me as one of the readers and thinkers.
I am one of those "meet me first" partners. I know it makes me a lot more comfortable, and I have been doing a lot of thinking about WHY it helps.
One thing - I think it's a lot harder to behave in a disrespectful or hurtful fashion to someone you've actually met. If you only know there is a partner out there in the abstract, it's much easier to disregard their feelings or needs. At least in my personal experience, when I've met my husband's other partners, things become a lot smoother for everyone.
The one I never got the chance to meet (it was cheating, before we opened the relationship) - well, she sent lots of emails telling him why he should divorce me and be with her. Peppered with an astonishing amount of criticism about what a terrible person I was - interesting, since she knows little or nothing about me. I think had I been a real person, rather than a paper doll, it would have been more difficult to behave so badly.
With the meetings, there's no interview or permission slip or anything like that - just a simple hello, so we both know we're real people who will be treated with respect. If they want to be my friend, great. If they are more comfortable with some distance, that's fine too. It depends on whether I click with them.
I see this not as a restriction against my husband, but a step to protect myself. I was recently burned, so I'm using potholders now. Is that really such a terrible thing?