You've hit the nail on the head with all your posts! Yes to correlation vs causation. There may be a few other things at play here.
I think it was you (forgive me if I'm wrong) who described in an earlier post a "walking epidural" where local anaesthetic type and dosage is adjusted to allow a woman to be mobile. It's not offered where I live (I'm not in Europe), but it may be more commonly used in France where epidural uptake rate is higher so protocols for walking epidurals are easier to establish. It can help with coordinating pushing etc if you can feel the contraction. Also mobility during labour tends to speed things up.
I also suspect that pelvic re-education must make one more aware of where on earth pelvic muscles are and what to do with them (for the next baby).
I live in a system, which like the NHS seems to discourage epidurals. It's cheaper. But the number of times, back when I covered labour ward, that you get called to an exhausted woman to put an epidural in is phenomenal. That mum is highly unlikely to push effectively when the time comes. You could usually predict who you'll be meeting later for the inevitable Caesarian. Yes, most of the time it is a funding issue and we are busy b/c we also cover theatres and we can't leave an anaesthetised patient, but I'm also aware that some midwives lie to their patients that "anaesthetist is busy" when they haven't actually checked.
It is sad that women continue to be short changed by dogma which is ultimately driven by trying to save money and staff continue to get harmed in the process by poorly designed working spaces and practices, again because it is cheaper.
The nitrous exposure is just the tip of the iceberg - lead gowns (and their inadequacy/ non existent checks on integrity) and radiation exposure, PPE practices and ventilation - the chickens are coming to roost unfortunately. Frankly I'm surprised my two DC were born at term and healthy, which is awful to think about. And no, I couldn't ask my employer to pull me off radiology lists and the like during pregnancy.