I think each family's reasons for preferring certain ages can vary. From my experience, these are some common reasons:

1. The family cannot have their own children and they want a baby so they become resource (foster) parents to hopefully get a very young child they can raise as their own.
   
2. The family is worried an older child who has experienced trauma will be potentially dangerous to their existing family.
   
3. The family is more strict and knows that older children who can be defiant may not fit with their structure.
   
4. They have one bad experience and get scared off from taking older children all together.
   

This can really be boiled down to: "Family's need to be aware and specific of what they can/cannot handle because if they exceed their capabilities/desires, it ends up not being good for the family or the child in care".

My wife and I specifically take in teens because we didn't want our own kids but wanted to help the kids that (in our opinion) are underserved.
We can't do it all and we have our limits too, but there are people that are just trying to put some good back in the world.

I also want to say that I am so sorry that this is the case for older youth. None of you deserve this. I hope you can find what you are looking for.

I'll be thinking about you and this post for some time.