I had a very similar experience. Induction, emergency c section and a GA a the epidural didn't work. DH didn't get to hold DD. I woke up to her next to me and had to ask the midwife to take her to DH as I was still coming round in recovery.
I always wanted to experience labour and birth, I even considered surrogacy before I met DH as I was sure I would never meet anyone to settle down with. So, to have a c section under GA was absolutely gutting to me.
For a few months after the birth I would cry everytime I saw a birth scene on TV or read birth stories. I felt awful my DH didn't get to witness the birth of DD and I often wondered if she was actually my daughter. What if the babies got mixed up? Sounds crazy but now, a few years down the line I'm absolutely OK with the birth. I obviously wish I had been conscious as, being our first, I've never witnessed a real life birth but I no longer dwell on it as much as I did in the early days.
DD is an absolute gem. I can honestly say that not having skin to skin has effected her in anyway. She's cuddly and loving and just an all round awesome kid. I did feel a bit detached from her for the first week or 2, as if I was babysitting someone else's baby, but I may have felt that if I had been awake when she was born - who knows!
Will this feeling pass pass, for me it did and I'm sure it will for you too. Although I do have this sense of never having given birth and so when I see or hear things about how strong woman are during child birth I don't feel that applies to me, but the feelings I felt about the birth initially have passed.
Don't be so hard on yourself. Your baby is here and healthy and doesn't care how they came into the world. It's early days for you, and hormones are all over the place. You are still finding your feet, there's lockdown, you are probably sleep deprived and you had a traumatic birth. On top of that you lost your first baby which is also probably playing a part of how you feel. There's a lot for you to be dealing with right now so be kind to yourself. You done what was best for your baby.
My advice is to give things time to settle down and see how you feel. If you are still feeling like this in a month or so then consider a birth debrief or speaking to someone about your experience. Feel free to PM me if you wish 