"so why does that logic apply to births?"
Because the research looks at outcomes by planned place of birth, not by actual place of birth, and only looks at mothers deemed low risk at the start of labour.
So - take 100 healthy, low risk women planning a hospital birth. Match them with 100 healthy, low risk women planning a home birth. Match them for age and social class and anything else which might possibly impact on their birth outcome.
Next step: compare their birth outcomes.
Out of these 100 women who'd planned a homebirth, who'd been categorised as 'low risk' at the start of labour, approximately 5 (say) would end up having an emergency c/s. Obviously they would have transferred into hospital to have this done.
Out of these 100 women who'd planned a hospital birth, who'd been categorised as 'low risk' at the start of labour, 10 (say) would have had an emergency c/s.
Neonatal outcomes the same for both groups.
So we say 'hospital birth and homebirth have similar neonatal outcomes, but hospital birth is associated with a doubling of the rate of emergency c/s'.
Simplified HUGELY of course!