And this despite the fact that higher-risk deliveries are more likely to occur in a hospital, so that if the environments were equally dangerous, you'd expect the stats to tilt the opposite direction.

Source? I've never heard that and it sounds like fear mongering. Homebirth, for example, is statistically very safe

How much of that is Simpson's Paradox, though? What I'd think of as 'homebirths' are typically informed, prepared, healthy mothers with no known complications...