I wouldn't go that far.

There are a lot of options for anaesthesia, and many of them are not opioids. For example, there is a whole set of halogenated ethers like sevoflurane and isoflurane (the most typical inhaled anesthetics). Their chemical structures and mechanisms of action are very different from opioids, so the vaccine-produced antibodies would not recognise or react with them. This is true for many of the other options (benzos, ketamine, even xenon gas).

That said, opioids are useful. They are very good analgesics (pain relievers) and can be used as anesthesia in certain situations, like cardiovascular patients. However, for most people in most situations, there are other non-opioid options.

Opiates != Anesthesia

Opiates are painkillers, anesthetic are a different beast.

Fentanyl is an extremely common ingredient in a surgical cocktail.

It is indeed, but as an analegesic, not an anasthetic. Paralytic, analegesic, anesthetic, is the usual trio for general anesthetic surgery. For “twilight sleep” as used in most colonoscopies, it will tend to be fentanyl and Versed.