Please don't be worried OP. I have not had a CS, but this topic has come up among my friends who have had only ELCS or EMCS. I had a vaginal birth, ok a bit scary/intense because it was fast, but not anything like a forceps situation etc.
This is what I tell them. I felt nothing for my son when he was born. Nothing. Nada. Zip. He could have been a kitten or a chicken or a bag of sausages, I did not give one shit about him, I was just relieved the whole thing was over. And this was all while he was having CPR as he inhaled some meconium when he was born. Even when he was on my chest afterwards, I felt nothing. Just spaced out shock, DH was the same, it was all a bit "errr ok now what?! Holy crap there's a baby here!". My friends who have had CS say they think it was their CS that made them feeling nothing/hard to bond immediately with their child - I said errrr no! It totally depends on the situation. It is normal to not feel a rush of love/feel much of anything. Some people do and some don't. I didn't feel true "love" for my son until he was about 4 weeks old, or older.
I think I would definitely have felt more for DS/bonded more had I had an ELCS - because I wouldn't have been in such shock! Can't say what it would've been like with an EMCS though, probably would've been just as shocked.
DMIL who had an EMCS under general anaesthetic bonded immediately with her DD, didn't understand what I meant when I said I felt nothing. So the "missing Christmas Day" analogy is quite false, and I think very bad from a mental health of the mother perspective.
I think I will feel more for my next baby (currently preg) at birth, because I've done it before. But I'm not expecting to feel a rush of love this time immediately either - because regardless of how your baby comes out, it's a big deal, and a lot for your head to deal with BEFORE you even get to the concept of the tiny human being that was the product of the situation.
I think it's totally unrealistic to expect some amazing "bonding" experience at birth, and fixate on a vaginal birth being the key to that. If it's your first child, you don't know what it is to love a child. You've never done it before! It's like falling in love with a man/woman romantically - some people feel love at first sight, others it takes time to build up and then they realise after a wee while that what they feel is love!