In my experience these forms are only signed by patients, so why are they valid in those cases?

The signatures per se don't really matter. What matters in contract law is if there was an actual agreement. The signatures are just evidence that there was an actual agreement. They are neither necessary nor sufficient. This is why verbal agreements can be binding contracts, it's just that these are harder to enforce because it's harder to show that a verbal agreement was actually entered into.

Well if the patient swapped the form, it would be pretty clear they didn't actually agree to the terms of the original form.

And if they swap the form in secret in order to trick surgeons into performing an operation which they haven't agreed to pay for, it's pretty clear that they're committing fraud.

Fraud is a strong claim, and it's not clear that there is even an implied contract formed - a contract requires defined consideration. An arbitrary amount conjured out of a hat after service is not well-defined consideration.

From my understanding, the medical billing system is so fucked precisely because hospitals bought laws making it so they can arbitrarily bill patients without forming a proper contract (otherwise there would be no way to charge someone who was admitted while unconscious). In the absence of such laws they'd have to fall back to an unjust enrichment claim, under which it would be pretty hard to justify charging someone $25 for an aspirin.

People keep saying “in secret” but that’d only be true if it happened after it were filed or some such. Giving someone an amended contract isn’t fraud.

Giving someone an amended contract isn't fraud, no. Giving someone an amended contract while making them think it hasn't been amended is a different matter, though.

What do you mean "in secret"? The hospital received the form in full detail. There is nothing secret about it. It's like claiming the ToS are secret because none reads it.

No, it's not at all like that. If someone hands you papers to sign and you give them different papers without telling them so, that is clear intent to deceive, which makes it fraud, or at minimum makes the contract void.

When I was registering as a new patient at a dentist, they gave me the standard form that would let them do things like contact my employer if I didn’t pay my bill on time. I stroked a bunch of that stuff out and initialed it.

If it ended up in court and they produced that form, would my amendments mean anything? Is it a contract when only one part signs it?