"Shelters do extensive behavioral profiling of each individual dog."
Please don't make blanket statements like that. I've donated training time and fostered for a few no kills. To say shelters do all that evaluation just not a true statement. Shelters have morphed into small business and are becoming used dog stores. Which is not a bad thing if dogs get a second chance. However, when money changes hands ethics get stretched. I've see too many 'Rescue' people weave too many heart wrenching fables to get dogs sold and too many times the wrong dog/wrong family matches happen.
"mixed breed can temper some of the behaviors and also improve the dogs health by eliminating some breed specific health issues."
That is only possible if both sire & damm back at least two generations are all free of genetic health problems. It's another 1/2 truth that is floating around the mutt community.