C-sections are scheduled in advance (sometimes over a month in advance) because the American medical system has optimized for cost reduction (which includes extreme risk aversion) rather than patient outcomes.
Additionally, there are many factors that can be used to determine the risk of natural birth.
However, there was a significant difference in the reasons things went wrong. Because in a hospital you're more likely to receive unnecessary treatment due to the "cascade of intervention"; and interventions such as C-sections have inherent risk. While at home, you're more likely to have an actual emergency.
Mostly the advice we got was, if you have a medical condition that increases risk, then go to a hospital or a birthing centre. But if not, and you live reasonably close to a hospital, home birth is quite safe.
I am speaking from personal experience with births in one of the biggest US cities in the last five years. Anecdotally, I think that some less-dense areas have better CNMW practices and more access to midwives due to traditionally less access to hospitals, but it seems very variable.
As a legacy of upright locomotion balanced with large cranium size and infant helplessness, humans have ended up with an objectively poor design for giving birth to offspring. Death during that process is “natural”.