There are about 9 church specific umbrellas in Harford County. I still remember when I began looking for an umbrella...I went to the state of Maryland's list of umbrellas and began scrolling through in alphabetical order trying to find one that my family could join. I called several which told me that their umbrella was open to church members only. They each said it a little differently, but that was the basic idea. No one explained to me the reasoning behind it and I immediately felt like an outsider. Still, I hoped I could find a place I could join. I wasn't sure if one was out there. Thankfully, I eventually did find one.
A few years ago, we became members of a church which is connected to a church that has a church specific umbrella. I began to ponder whether or not we should join the umbrella or if we should remain in the umbrella we were in. I had several conversations with people in the church about it that helped me understand the reasoning behind church membership being a requirement for membership in the church's umbrella.
If you are a member of a church, you have placed yourself under the authority of the elders of that church. You have the same doctrinal beliefs as the church (or agree to submit to them). There can be a blessing in that accountability because the elders are there to shepherd and encourage your family.
When an umbrella is church specific, all the members of the umbrella are also members of the church. So, the families fall under the authority of the pastor and elders of the church. If there are issues, the elders can lend support. When you are members of the same church, there often is regular weekly contact because you see each other each week at church and youth group. Also, because the families hold the same doctrinal beliefs there is a common set of values which gives the families a base of commonalities, especially when it comes to what you are teaching your children about the Bible.
As a member, you have also committed to caring about and supporting the other families in your church. One of the challenges that many homeschooling groups face is taking care of everyone. Some churches set limits on how many families from that church can be in the umbrella and I have known several families who were unable to join the umbrella at the church they belonged to. Belonging to a church that has an umbrella is not a guarantee that you will be able to be a part of that umbrella or will be included. One umbrella locally formerly included families outside the church, but became a church specific umbrella because they felt they weren't able to take care of the families within their church who wished to join. Church umbrellas are church specific so that they can care for the members of their church.
Nonchurch specific umbrellas do hold families accountable as well. The amount of contact with each family varies based on the needs of the family. As I've now been in both types of groups--a church specific group and a non-church specific umbrella, I've realized that both hold families accountable--and one not more than the other. It is convenient for church members to review families in their church and have regular ongoing contact, but it is also complicated because they are often friends, even good friends. On the other hand, reviewers in nonchurch specific umbrellas have to make more efforts to be in contact with the families they review since it doesn't naturally happen at church each week, but they are able to step back and potentially be more objective and helpful in that way. One of the non-church specific umbrellas in the area has several reviewers who's children are all grown--but these women continue to review families as an encouragement to those who are homeschooling.
I hope that this explanation is helpful.