Are there specific agencies that match birthparents of older kids and couples?

Your post is old, sorry this is a belated reply. I think you have it backwards--it is not foster care experience that is bad for the children and causes issues, it is the reasons that led to foster care. Children in the system who are adoptable have more than likely had at least some time in a loving, functional home and the advantages of therapeutic services and medical care for the issues that led to placement. Those who are not, wouldn't have those advantages. If you truly want an older child, I think you should take another look at fost-adopt, and research the facts, not unfounded myths about foster care. I am a foster mom for a relative, but have gotten to "know" many fmoms through these boards. It is a miraculous thing, IMO, that so many people do so much for these kids for nothing in return.