Former Foster Parent here.
When I went through the process I was looking to adopt. The Agency is contracted by the state. They have to follow the state's rules, regs, and procedures. I dealt with two different agencies, one Lutheran and one non-religious. The only thing that let me know they were religious was their name "Lutherbrook". That and the Lutheran Church next door to "training" building and "residence" building. (Training was for Foster and Adoptive parent to take a variety of classes, and the residence was what I would guess was a battered women's shelter, as I never saw any men there, just women as some children.)
The only discussion we ever had in our training was about respecting the religious rights of the child and the parents, if their parental rights had not been terminated. So for example, Little Timmy's parents are Baptist and you are Catholic. You should make every effort to see that Little Timmy gets to Baptist services. It doesn't mean you can't take him to Catholic services, but he should be going to just as many Baptist services as Catholic, so that generally means you do one one Sunday and the other the next. However, if they are say Muslim and you as a infidel are not allowed in the Mosque, then you are not required to leave Little Timmy alone with strangers, in fact you are prohibited from doing that.
And of course, parents come up with goofy religions and "requirements" that you don't have abide by. There was nothing about "Only Jewish kids can be placed with Jewish Families", however they often did try to "match up" as best they could. The only group with an actual legal requirement was Hispanic kids, I can't remember if they had to placed with Hispanic parents, or just "should" be if ones are available.
I got out of because:
1. They would call you and say they have a placement. Which means they either got a new case and needed a home for this/these kids, or their previous placement had gone pear shaped and they were sent to a mental hospital, and were now "better" (Narrator: They weren't "better") and ready to go into a new home. The two I had came from the later and in both cases things went pear shaped, police were called and kids taken out of my home at my request. And they give you no information about the kid other then name and age. They usually are sitting there by the desk of the case/social worker, so they just want to get this kid into a foster home, so they can go home. So unless you know to ask lots and lots of question, all you get is happy talk about what a great kid they and they like sports and school and will be a really good fit in your home. Bull.
2. Unlike other states, Illinois doesn't terminate parental rights (For 230RN; aka TPR. ) unless and until a "Stable" placement is made. That is generally a couple years of no incidents, like mentioned above. Then they begin the legal process to TPR, once that's done, then you can begin adoption proceedings. The average from Placement to Adoption is about 5-7 years. One family managed to do it two years, because the newborn they initially fostered had been dropped off at a fire station. So, no parents to TPR.
3. Since Mom and/or Dad are still in the picture you have to do supervised visits, and in the day and age of electronic and social media are in fairly constant contact with their kids. And Mom and/or Dad generally enjoy nothing more then undercutting the Foster parents. Ask me how I know.
4. Yes, you can do private adoptions, but that is generally for newborns, where the parents contracts with the birth mother. All other kids are by law "Children in Care" (They changed the law, so they are no longer "Wards of the State".) And yes, there are "Orphanages", but they are not called that. They are called "Residential Care Centers".
Now, the reason I heard that Lutherbrook was getting out of the Foster care business was that the State of Illinois was more than 2 years behind on their bills. That's a lot of salaries and electricity to front the state, so they called it quits. I had not heard that the state fired all of them. The reason they had non-DCFS (230RN; Dept. of Children and Family Services) agencies was there simply wasn't enough state employees/social and case workers to managed the number of kids in care. I know Children's Home and Aid (a secular agency) had over 300 kids in just one of their residence facilities on the north side of Chicago. But I gave up my license about a year ago, so I'm not in that loop any more.
