I read an article by a historical linguist not so long ago claiming that by the time Shakespeare used they/them as a singular indefinite pronoun, the usage had already existed in English for centuries.

In fact, claimed the linguist, the usage dates back to the very origins of the language. The pronouns "they" and "them" came into Old English from Old Norse, brought across the North Sea by Vikings and Norse settlers. Apparently, from the very first use of the new pronouns, a singular indefinite sense existed. Just like the examples you wrote about. The linguist claims this is well documented.

In any case, writers have been using the pronouns in that sense since the time of at least Chaucer.

They/them as singular is built into English and feels perfectly natural to native speakers, who don't even think about it before using it as singular in an indefinite sense. (Meaning gender is unknown or not specified.)

Outraged traditionalists citing grammar don't know what they're talking about. Or perhaps they don't care. Conservatives often urge prescriptivist forms of grammar that attempt to cement unnatural "rules" on ordinary English usage, like not ending sentences with prepositions.

Prescriptivism is something up with I shall not put ... just speaking as a writer.