If you don't have epidural during labour do you have to be put to sleep for c-section?

F
I met with an anaesthesiologist this morning at my hospital to make sure that I would have the option of an epidural when I go into labour as my midwife was unsure because I have scoliosis (curved spine). The anaesthesiologist said my scoliosis shouldn't be a problem with me getting an epidural so long as I tell the midwives as soon as I get to the hospital that I want the epidural right away. Apparently it's easier to do (because it's all about making their job easier when I'm the one in labour, right? 🙄) and more effective when you get it earlier in labour rather than later?
He also said I should get an epidural because if I don't and I wind up needing a c-section I'll have to go under general anaesthetic and be put to sleep, which would mean that I would miss my own child's birth. What the?! Surely that's not right and he's just trying to scare me into getting an epidural instead of trying to go naturally!
So mamas who wanted to give birth naturally, but at the last minute for whatever reason that couldn't happen and you needed a c-section, were you put to sleep for it? Or were they able to give you an epidural right before the c-section??