I was arguing with someone the other day, and they said that bananas are too expensive.

I would and did flinch at that. It's really a generational thing. In my day (I'm 78), we would say "I was talking to a guy/woman yesterday, and he/she said that bananas were expensive." The assumption that the gender is irrelevant is actually political. Maybe it matters, maybe it doesn't. But it's information, it's available, and it costs no more is syllables to include. So why de-sex it?

Do you really want to cite Shakespeare's taking poetic license for grammar?

The singular "they" may well replace sexed pronouns. Context is usually enough to sort out the referent. But using preferred pronouns completely defeats the purpose of pronouns, which is to fill a grammatical hole without thinking about what goes in it. If you need to remember a third party's preference - someone who may not even be present - the whole point of that pronoun, as a PRO-noun, is lost.