Much discussion of language, particularly on the web, goes nowhere because those involved are very often talking about different things. In considering approaches to language, as in all else, it’s important to be clear what we’re talking about. In particular, the word ‘grammar’ is bandied around without much thought as to what it might be.
Some have commandeered the word to cast a cloak of spurious respectability over their prejudices and ignorance. Nevile Gwynne (here and here) is one such. It is true that the Oxford English Dictionary’s third definition of ‘grammar’ is
An individual’s manner of using grammatical forms; speech or writing judged as good or bad according as it conforms to or violates grammatical rules; also speech or writing that is correct according to those rules.
It is certainly used in that way, but ‘an individual’s manner’, ‘judged as good or bad’ and ‘speech or writing that is correct according to those rules’ all imply a degree of subjectivity, whereas it’s more helpful to think of grammar as a matter of objective fact. Specifically, the rules of grammar tell us how speakers of a language put together units of meaning to make words, and how they put words together to make sentences. 600 years ago, John Colet had much the same idea about Latin:
In the beginning men spake not Latin because such rules were made, but, contrariwise, because men spake Latin the rules were made. That is to say, Latin speech was before the rules, and not the rules before the Latin speech.
There is no evidence, for example, to support a rule of English grammar that says they can only refer to a plural antecedent. By contrast, there is plenty of evidence to support the rule that the past tense of regular verbs is formed by the addition of -ed to the plain form. What linguists specialising in grammar do is search for such evidence and codify the results.
Style is quite a different matter, one not of fact, but of opinion. Some may say that they referring to a singular antecedent is awkward or ugly or illogical or whatever you like. They may conclude that for those reasons its use makes an utterance ineffective. Others may take an opposite view. What neither can do say is that it is ungrammatical, when there is so much evidence for its use in the prose of reputable writers over the centuries.
It’s like looking at someone’s shirt or wallpaper. You may not like the colour of the shirt or the pattern of the wallpaper, and you’re quite entitled not to like them. But you can’t deny that what you’re looking is a shirt or wallpaper.