I am absolutely not downplaying those instances when something goes quickly wrong. What you don't hear so much is the stories of when something goes wrong in the hospital where there is a negative outcome but importantly where it wouldn't have made the blindest bit of difference where the mum and baby were, there would still have been a negative outcome. So if they'd been at home it would have been blamed on 'homebirth' whereas if they are in hospital it would be 'one of those things'.
One thing I forgot to say is that 'labour isn't progressing' is often a main reason for transferring in in labour, as is that the woman wishes to go in (changes her mind and this IS ok to do) or that the woman is requesting further pain relief (again totally fine). Labour not progressing as expected is a big reason why interventions happen, and yes if you stayed at home something might happen to you and your baby and that's why the midwives say to go in at that point so really it's a moot point.
Yes, in Africa people have births at home and more women die but that's not because they are at home, per se, it's a combination of poor nutrition and maternal growth, lack of antenatal and intrapartum care and risk of infection. It's not really a like for like scenario.
The trouble with all of the things is that no, you can't go back and make a different choice and the risks are small but the stakes are high and that's why people are so passionate about this. But it isn't as simple as 'get everyone into hospital ASAP'; we know (from research) that being in hospital, especially on non-indicated continuous monitoring, increases your risk of unnecessary interventions. So ideally we'd want to keep things as low risk as possible, while not interfering where not necessary, but with easy reach of medical assistance if things do go wrong. The ideal I guess...
The most important thing I think is that people choose where they have their baby based on where they are going to feel safest. And not feeling like they have to choose a home birth because they are so scared of hospital or feeling they need to go into hospital early because they are so terrified of labour at all.