I am sorry, but that is simply a terrible idea. There are millions more orphans worldwide than there are families willing to adopt. It is extremely rare for an orphan to have more than one family to ever attempt to adopt them. Moreover, in many countries, orphans age out of the system fairly early on. For example, at 14, a kid in China is no longer eligible. So why on earth would you allow them to be listed with an agency that would screen out otherwise qualified families simply based upon their religious beliefs or sexual orientation, when that might be that kid's only chance at a family. Its flat out unconscionable.
We adopted special needs twice from China. We went through a Christian agency. We didn't choose that agency because they were a Christian agency, but rather because they were reputable agency that happened to be Christian and they were the ones that had the girls we were interested in adopting. My wife and I are not religious. So had they refused to allow us to adopt because we were not religious, then our girls would still be in China. One of our daughters has Cerebral Palsy. China is not like the USA. There is no safetynet there. They don't do a good job of taking care of those with disabilities. So had we not adopted her, chances are no one would have (way more orphans than families willing to adopt). Thus she would have spent her childhood in a ****hole "social welfare institute" in China where she would have received little to no education, no physical therapy, and malnourished. Once she became an adult, she could look forward to life on the streets as a beggar. Moreover, she would be branded with the surname of Yu, so that anyone in China would know she grew up in an orphanage without any education.
That didn't happen of course because our religion was not part of the criteria for our being approved to adopt, but rather it was what kind of a home we had, our parenting ability, the stability of our home, our finances, our background and so on. Thus have we 3 kids now, 1 biological and two adopted girls, a good, loving and stable home, they all do well in school, are involved in all sorts of activities, and almost certainly grow up to be good productive members of American society.
So that's why its a terrible idea to just allow the bigotry on the part of an agency to play in role in home studies or adoption approvals. We are literally talking about the lives of kids here.