In reality, there is only 1 argument that has some merit to it against they/ them.

It always fascinates me when someone with a strong view on a subject can only think of one argument against their position, an argument that just happens not to hold water.

The best argument against "they/their" as singular pronouns for some people is that they require us to know the pronouns of all people. That is not how language works. We cannot be obliged to know the pronouns of each and every person with whom we interact.

Sure, people can tell us their pronouns, but only some sort of elitist solipsism would lead someone to think we are going to remember their pronouns when we are often not even able to remember their name. Demi Lovato will be "she" to anyone who sees DL but does not know DL. If DL's friends want to do DL the favor of remembering DL's pronouns, that's their privilege. But I've got enough to remember...

The best argument against gendered singular pronouns is that they are unnecessary. The problem is not that "their" is wrong, but that "he" and "she" may be obsolete. Non-binary people are better off campaigning for uniformly applicable third-person singular pronouns to replace the gendered ones than trying to get people to remember who is what.