I was also going to speak about the descriptive and prescriptive approaches to languages, and then dropped my paragraphs because it was not clear to me whether OP acknowledged that singular they was a thing in English. I decided to consider that they effectively acknowledged singular they.
By taking a prescriptive approach (telling how people should speak English), maybe singular they can be deemed invalid in English… depending on who is prescribing the usage. But people taking a prescriptive approach do not agree between each others, and even if they did, they would never represent the wide, heterogeneous set of English-speaking people.
Case in point: they do not agree here, but the usage of singular they seems to be more and more recognized / recommended [1]. A descriptive approach will acknowledge a widespread use of anything, regardless of the reasons why things are used.
About politics and languages. Politics are constantly shaping the language. This is not new. It's part of the natural evolution of languages. Politics are not inherently bad neither. We don't have slaves anymore in most countries, thanks to "politics".
Actually, I find politics is actually too vague a word, and some people seem to invoke it as a bad thing, saying that we should focus on technical matters when their view opposes the view of people seeing something belonging to the status quo as an issue (which is a political stance in itself… and considering that technical matters and politics are opposed too, ironically).
You can't really reject an idea "because politics". You may either argue, or disagree respectfully, but there is no such thing as avoiding politics and remaining technical.
[1] https://en.wikipedia.org/wiki/Singular_they#Acceptability_an...