I had my first two babies at the Lindo. A couple of things: it is well known for being very hot on the more medical side of things. Those who wanted a midwife led private birth tended to go either to John and Lizzies or the Portland. I was paying for top notch medical care, and that is what I had there.
The other thing is, very few first births go anywhere near to plan. Nothing, even 17 grand, will guarantee you have the birth you want. In my experience - - three babies, three NCT cohorts, three sets of baby groups, toddler groups, three school years - - nearly every first birth has an element of shock, fear at times, perhaps disappointment too, and some much more serious injury. You cannot throw money at it and guarantee a peaceful, easy birth. You can throw money at it and expect a midwife there when you need her and a consultant who knows the evidence base and the pros and cons of every intervention. And a private room and a clean loo.. If you have complaints, do bring them to the hospital as I am certain they would want to act on them.
I was very glad I had my first two privately as I was very nervous and absolutely needed the hand holding. It was a traumatic birth, 10 hours, ending in a section. Had no wider family support etc Stayed in five days and learned about baby care there. By dc3 we had moved away from London, to an area where there are no private hospitals, and I had dc3 on the NHS. The antenatal and postnatal care seemed bare bones in comparison but the birth experience was just as good. Mind you, the first was an EMCS and the others were elective sections.
A long way of saying that what you describe is quite well known about the Lindo approach. But it is not about squeezing in as many patients as possible, it is about the evidence base and what is safest in particular circumstances. And they are very conservative on that, and known to be so. Their rate of C sections is comparatively high, but their rate of birth injuries to mother or baby, is miniscule (or was when I was researching this.) For me those were the key elements I was paying for. If you were keen for something different, there are other choices and not sure why you would pick the Lindo which, as I said, is really well known for this approach.
I hope you are on the way to recovery and enjoying your baby💐