Totally normal fear to have! Even people without HA have it pretty often. But the odds of something going wrong are slim, and even if it does, you're surrounded by people trained in how to deal with it.
To give you an idea, my friend's dad had some pretty serious hernia repair surgery. The guy has heart problems, etc, he's an elderly man far from in the best shape of his life, so keep that in mind. He did stop breathing at one point during the surgery.
When my friend's mother was being given the report on how he was, the surgeon said something like "we ran into a bit of trouble when he stopped breathing, but he's fine now" like he was saying "oh, it's raining out, bring an umbrella."
So even when things that sound horrific to those of us with anxiety, to them it's just a little bump in the road. They don't go to school for a decade or more for nothing!