That's what a pronoun does. A pronoun is a word that we substitute for a noun or noun phrase. They are substituting a pronoun (they) for a singular noun (the waiter), thus using the word they as a singular pronoun.

Yes. Which is exactly what I said. "They" CAN be singular. It's flaw as a universal pronoun is that it is not INHERENTLY singular. It can refer to multiple people and can refer to anyone.

It is also breaks a frequent use of pronouns because if you use they, you can also use he and she and be referring to the same person. Take a simple sentence like:

"They were arguing last night, he said she doesn't appreciate him and she said he should leave her then". Utterly unambiguous. Try writing that in a gender-neutral way "They were arguing last night, they said they doesn't appreciate them and they said they should leave them then"

Aside from the fact that literally no one could understand that new sentence, it also doesn't work. My grammar check is currently angrily telling me that it is "she doesn't" but "they don't". There are places where you cannot substitute "they" in properly.

This is especially a problem because people tell stories without identifiers. You might say "so I was talking to this guy and this girl" and never use anything but pronouns to refer to them ever again. But if you use "they" as a singular in there to refer to one of them, it breaks the entire ability to use pronouns, because someone could be he or she and still be they. You wind up saying "the guy" and "the girl" in every sentence.

They CAN be used as a singular, gender-neutral pronoun. That doesn't change the fact that it is a terrible choice for one.