What is the #1 defense to a birth asphyxia case?

Very simple: the most common defense to a birth asphyxia case is "causation".

Causation means that something other than the traumatic birth caused the baby's brain damage.  The defense "causation" arguments include congenital abnormalities (genetics), lack of fetal movement or growth during pregnancy, and injury to the brain after birth, but the most common is based on premature birth. Congenital abnormalities can be ruled out with genetic testing and abnormalities during pregnancy can be ruled out with a normal prenatal medical record. However, babies born premature (less than 35 weeks of gestation) often have multiple health issues because they are born before their organs have fully developed.

If  your baby was not premature, genetic testing rules out congenital abnormalities, your prenatal medical record is normal, and your baby did not sustain any injuries to his/her head after birth, you are well on the road to proving that none of those factors caused your baby's brain damage.  This is how you disprove the defense's causation argument.