Intra country results can be challenging as who chooses a home birth & who goes to the hospital is already a likely biased population. Maybe the pregnancy is already known to be an at-risk factor, thus biasing things. Or choosing one vs the other may be a confounding socioeconomic factor.

If it's this hard for intra country comparisons, why would one expect inter country comparisons, where one country has socialized medicine & the other doesn't, to meaningfully reflect the specific policies around child birth vs the larger differences between societies?

I don't know the answer, but I don't think simply comparing mortality rates can be used to conclude "natural home birth is safer".