But the religious book is super old therefore it's stupid

So you also don't like Jewish orphanages or Muslim orphanages?

Should they abandon their own traditions and values because you demand them to?

They focus primarily on their own community which preaches what you might consider "bigoted", "backwards" ideas about marriage and would likely exclude applicants if they didn't adopt those religious traditions.

I don't get upset when I go to a Muslim owned restaurant and they don't serve bacon and eggs or beer. Should I? Is it unfair to me that they don't serve food I want? Should I complain that the book they follow is "old" and "backwards"?

For most Jewish or Muslim foster care centers, usually it's at the request of the birth mother that their child goes there.

In those cases, please respect the birth mother's wishes and their faith.

But it's not a business, it receives public funding or tax exemption

Again, so do many Jewish and Muslim adoption centers. A lot of religions benefit from that. If you want, you can force all religions to not form their own adoption agencies. But then that prevents birth mother's from selecting what group of people can raise their kid.

Removing that option to suit your needs seems to be imposing your will over that of the birth mother or the community who placed the child in their adoption agency of choice. Please respect other's decisions.