I wouldn't try to compare the pain from a medical procedure to deal with a medical problem to birth. It's completely different. I've had 2 reconstructive knee operations and I had severe pneumonia which left me with fluid around my lung and in horrible pain every time I took a breath. Those were painful!! Giving birth was nothing at all like that. There is pain and discomfort (honestly, for me, more discomfort than pain), but it's fleeting, 30-60 seconds every few minutes. It's not constant pain like what you're experiencing and it's largely short-lived. The average first labour is about 9 hours. So imagine if you felt some of the pain your feeling now but only for at most a minute at a time, with a nice break after when you felt completely comfortable, and only for the first 9 hours and then you felt fine again. It's more like that. The constant, chronic pain from an operation or injury is completely different and I think much harder to deal with. I didn't use any pain relief when I was giving birth other than a TENS machine and the only time I'd say I felt "pain" like what I felt when I had my operations or when I had pneumonia was when they did a vaginal exam (that hurt!) or when I got stitches. The rest of it was really manageable. And again, I didn't use any sort of pharmaceutical pain relief. I was in labour for 12 hours (so longer than after for a first baby) and I pushed for 4 hours (again, longer than average!).
Now, yes, if you have a c-section, you will be in that sort of pain as if from an operation, but only about 10% of birthing women actually need a c-section (even though the rates in some places are as high as 35% or more, they are largely unnecessary or elective ones), so just make a real effort to reduce your risk of needing one - avoid an epidural unless you need it, be upright, don't be induced unless it's for a true medical reason, etc. You'll manage, with support from your partner or family, just like lots of women do. But I wouldn't stress about that now as you more than likely won't have to worry about it.