Yes it is cold in theatre. I work in obstetric theatres, so lots of experience. But skin to skin is absolutely a thing for C-sections, especially electives.
As others have stated, your gown can be left undone at the back. Which makes it effectively a thin sheet with sleeves. Very easy to lift up and plonk a baby on your chest then cover over. You can make even more room, by leaving an arm out of one sleeve. Monitoring leads can be placed on the back of your shoulders and your side to help with access as well. The drapes are plenty loose, so easy to tuck a baby under. As for the drips, you can work around them. They should all be going through one cannula anyway.
Then there's these amazing innovations called towels and blankets. Your midwife obviously hasn't heard of them. Very effective at trapping heat. They can even be pre-warmed ahead of time on the baby warmer, though some theatres may have a warming cupboard. Then she could occasionally check baby's temperature with a thermometer to ensure baby isn't getting cold. As an aside, if your requests are ignored, and your DH/P is feeling confident, once he's been handed the baby he can unwrap, place on you, and re-cover. It's your baby.
You asked about cutting the cord while baby is on your chest. Pretty impossible, as umbilical cords are rarely long enough to go over the drapes and keep everything sterile for the surgeons. Delayed cord clamping is pretty standard nowadays anyway, with the proviso that you and baby are well ofc. Baby will be down on your legs for a minute or two, then the cord is cut. Some surgeons also milk the cord into baby before cutting. If you ask, and they're kind, they might also leave it long, so your birth partner can trim it down if you'd like that.
I didn't get skin to skin with mine, which I am slightly salty about. I don't remember being asked, but then I didn't ask either. It was a 2am emergency though, and my memories are very hazy about the whole thing. I probably looked like I couldn't hold a baby. DH was delighted to get the holding time in though, which I cannot fault him for.
So... Yeah. I honestly can't see any reason why not, other than a hospital's backwards policy.