I don't want to scare you, but I also swallowed all the NCT BS and was anti-induction because of the 'cascade of intervention'. Unfortunately I was unlucky enough to have an ob consultant who was very laissez-faire. I trusted her, thought I was being monitored (didn't get any scans/doppler scans/monitoring which I now know I should have had).
I ended up having an emcs at 42+3, 30 hours after they finally did a scan at 16 days overdue which revealed the amniotic fluid level was down to zero. Literally no fluid left at all around the baby, just a tiny pocket in front of the head which is what they could feel when they did the sweeps, and only because the baby was only 2/5 engaged. When it finally came, the c/s was due to fetal distress - heart rate plummeted because of cord compression. DD was born with plagiocephaly (basically her head had been jammed into my pelvis with no protective fluid, causing one side of her face to be squashed, one eye to bulge out and not close properly, and the other to be pushed in and barely open) and torticollis. It took a year of physio to heal the neck muscles and allow her to turn her head fully and freely. She also has severe reflux which is apparently more common in post-term babies due to insufficient fluid/swallowing of meconium, etc, though there could be other causes.
In my case, induction wouldn't/didn't help - I was already having contractions every 3-4 mins when admitted, but they decided to induce anyway. Just meant that they put stronger pressure on DD. Nothing happened because she was well and truly wedged but they appear not to have considered that option (head turned out to be 98th centile though she was below average weight even for term, due to placental insufficiency - it had failed and started to degenerate to the point they couldn't actually remove it, it just fell apart inside me).
We were really lucky, we found good therapists/paediatric consultants (mix of private and NHS), helped by our excellent GP, and after 6 months of repositioning DD every 45 minutes during the night (yes, I had a stopwatch going off) and a couple of hours of physio exercises a day, the plagio improved so it's nowhere near as noticeable and after the year of physio the torticollis was resolved. She still has severe reflux at 2.
Anyway, I didn't want to scare you but you asked the question whether it really is dangerous to wait. Yes, it can be. I was naive. I never, ever want to be in that position again, and I wouldn't wish it on my worst enemy. I honestly thought we'd lost the baby when I was lying in operating theatre and was told they didn't have time to wait for my husband and didn't have time for the anaesthetic to take effect before starting. It never occurred to me you could get past 42 weeks and then not be able to wait an extra 2 minutes for anaesthetic. It's EXTREMELY unlikely you'd have the same circumstances (unless you're at the same shit hospital I was), but one thing the NCT teacher never told us was that there's a downside to waiting, and actually sometimes there's a reason that baby's not coming out. I'd definitely be pushing for full monitoring/scanning before induction though, just in case. I'd rather go straight for the c/s if that's the way it's going to end up anyway. If fluid levels are already low, they should not induce, it's too stressful for the baby.
Sorry this is so long but I was in exactly the same position as you and I really wish I'd known then what I now know. I still wake with flashbacks every night, and we were lucky with the outcome.