I must admit my bias up front. I am a pediatrician and an active member of the medical community in Colorado.
Giving birth outside of a hospital where emergency services are available 24/7 is a needless risk, and a significant one at that. Risks to the mother (and they are significant) aside, lets think about what can and does happen to the child and the timeframes involved using two common complications: placental abruption, and compromise of the umbilical cord. Both occur in otherwise normal pregancies, giving no warning no matter how good the prenatal care may have been. An overwhelming majority occur during labor and result in compromise or complete loss of blood supply to the infant. This is equivalent to the child suddenly not breathing. The infant has a similar tolerance to this insult as anyone else does, and will die within minutes if no intervention is made, just like anyone else.
The intervention I mention is IMMEDIATE delivery of the child, by "crash" C-section if needed, followed by CPR. The only way to know this is happening and the only way to intervene is to have fetal monitoring during labor and the ability to perform a C-section and resuscitate an infant while caring for the mother. Minutes count. Even being "close" to a hospital would realistically take 20-30 minutes for travel and triage, not to mention starting to help the mother and child. If something catastrophic has occured, that is far too long.
I may sound alarmist. I assure you I am not. 1 in 10 normal deliveries require some form of resuscitation beyond stimulation, warming, and oxygen. 1 in 100 normal deliveries require intensive resuscitation with artificial respiration and chest compressions. Luckily, our success rate for recovering the infant is quite high, but these are not numbers to take lightly.