I would expect it to be uncommon for keepers of pit vipers and corals in the US to stock antivenom, but I'd expect it to be common for keepers of non-native snakes because US hospitals don't stock non-native antivenoms. You'd typically have to obtain it from a zoo that keeps exotic snakes.

Antivenom is never "worse than the bite itself." It can occasionally cause an anaphylactic reaction which is easily manageable in a hospital setting by administration of epinephrine. The reasons doctors hold back on antivenom is that a significant fraction of people come in to the ER thinking they've been bitten by a venomous snake when they haven't, or they've received a dry bite from a venomous snake. Antivenom is never administered until venom symptoms are evident because it's quite expensive and there's no reason to risk a (manageable) anaphylactic reaction if it's not necessary.

Withholding snake* antivenom once symptoms are evident would often be a death sentence, and always medical malpractice.

https://www.sciencedirect.com/topics/neuroscience/antivenom

*Edit: Added the word "snake" because black widow antivenom exists but BW bites are not usually life-threatening in adults. Most people can pull through without it. This is much more risky with venomous snakes.

A zoo would be a centralized source. Depending on the country, there are central hotlines to help doctors know what they need and where they can get it.

An anaphylactic reaction on top of a snake bite can be worse than just a snake bite, but of course it depends on the circumstances and species. Also the immune reactions from the antivenom may clot the kidneys, which are probably already a concern if the patient is in shock. And yes, all of this is manageable. Immunology can be really weird and complicated...