I had a straightforward c-section. Bub came out, went to the paediatrician for a minute, and then was wrapped in a blanket and placed on my chest. I could hold him and talk to him while they stitched me up. And he just looked at DH and I as if to say "who on earth are you?".
You need to remember that having an anaesthetic also screws up Mum's ability to maintain temperature a bit, so it's probably not a great idea to rely on that to keep bub stable. And the theatres are chilly!