I had something similar to this with my most recent birth. Because the baby had anomalies, it wasn't perfect, but it was still so different than my first two c-sections. I got to hold Psalm in the OR on the table for a long time, and take her with me into recovery. And my midwife (a CNM) was able to scrub in and act as my nurse, so I still had that good support. Here they call them family centered c-sections (which I think is more accurate a label than "natural"), and they are actually now the norm at the hospital at which I deliver. (St. Luke's Baptist)
This was the norm when and where I had my second c-section 6.5 years ago. He was on my chest while they were still closing my up and nursing as soon as they wheeled me back to my room (under 30 minutes). I hope that we can do this again next year. For my last son, I had planned a VBAC but developed pre-eclampsia at 35 weeks and decided no induction. Since I have had 2 sections and a D&C for a miscarriage, I plan to have a third section.