YANBU - men and women are not the same, and don't see the things the same. There was some research that showed that men only invested in friendships with women, who they found sexually attractive.
Obviously that study is somewhat flawed, and I too have had male friends who never seemed interested, but when I was single, a lot of attached men made passes at me... quite a few of them were married to various friends of mine.
I don't know how things will pan out with DH & I, but I do know that if he was to make a female friend, and ever wanted to go for a drink - or anywhere alone - with her, I would automatically take it as they were going on a date. He had a female friend in the past, one he'd previously slept with - and my intuition was spot on - she still wanted him, and he couldn't bloody see it.
Old female friends - fine, but new ones, that are not a mutual friend, no - I would hate it.
I respect others have different views, and each relationship is different & has different boundaries.
But given the fact that he previously fancied her, I would take him going for a drink with her, as him going on a date. Anything less than not speaking to her again (except in a professional capacity) I wouldn't allow.
I also wouldn't invite her over either OP, I don't agree that you should try to befriend her. The problem (as I see it) is if he fancied her, then he still might have the flames smouldering. The more time he spends with her - including at your house with you present - the more he bonds with her, the more likely he might fall for her. She may not reciprocate, but instead of getting the hots for someone else, he should be investing in you.