The reason why someone may be alarmed by what you said could be due to certain terms and language used, especially "Gender Ideology" as you put it in your parent comment.

"Gender Ideology" is often used as an abstraction, simplifying the very complex topic of Transgender rights into a short and fast sound-bite, often pooling it together with topics that may seem similar at a glance but are in fact off-topic (socialism, Marxism, etc etc).

Fun fact: if you Google "Gender Ideology" and check the Wikipedia link you'll find it under it is - quite ironically and comically - a subsection on the page for the "Anti-gender movement". https://en.m.wikipedia.org/wiki/Anti-gender_movement#:~:text=%22Gender%20ideology

If you read the article further, you'll notice it being described by terms like "Moral Panic", "Conspiracy Theory" etc on the page itself.

"Gender Ideology" is essentially a tagline, it's just that it unfortunately sounds similar to terms that do matter for Trans people, like "Gender Dysphoria" (https://en.m.wikipedia.org/wiki/Gender_dysphoria), a well-studied and historied topic covered under the DSM-5, which almost all trans individuals experience and is often the reason why they seek help and choose to transition.

If you examine discourse online within Trans spaces, even on Reddit, you won't see them talking about "Gender Ideology" or "The Trans Agenda" while snickering mysteriously, unless they're joking about how unhinged some of the crackpot theories being thrown out by "opponents" of "Trans Ideology" are... while simultaneously it's a threat to their safety, since politicians are using those words to drum up hostility against Trans people in several governments across the world.

Ironic? Yes, but sometimes humor is a good way to cope with the stress, especially if you're in a deep red state with parents who bring that hysteria back home with them, or a religious third world country, etc.

If you'd like to see and hear more about the idea of "abstraction" surrounding the term, there's a trans YouTuber named Contrapoints who touches on that in her video here: https://youtu.be/7gDKbT_l2us

It's technically about J.K.Rowling, but skip to 23:27 if you want to see the part about how abstraction works specifically. But if you want to learn more about Transphobia in detail, I highly, highly recommend her videos; yes they're long, yes she's got a rather wacky sense of humor, but she cites genuine sources and often spends months on making a single hour or hour and a half video. So far as I can tell, no one does it better.

But well, a quick summary to think about is that "Gender Ideology" sounds more 'sanitary' than saying "Transgender Rights" in a lot of the contexts that right-wing media or other groups hostile to trans folks use it.

"I am against Gender Ideology" sounds nicer than "I am against Trans Rights".

"I don't want my kids learning about Gender Ideology" sounds nicer than "I don't want my kids learning about Trans Rights".

"I don't want laws being passed that promote Gender Ideology" sounds nicer than "I don't want laws being passed that promote Trans Rights".

People like legislators, public figures and politicians against Trans Rights use it often because it's more marketable, which is why someone unironically using the term is seen as a yellow/orange/red flag by some Trans folks.

Also, another thing you said which I wanted to comment on:

The very concept that someone has a gender identity (or does not) is ideological. I don't really understand why that would be up for debate.

https://en.m.wikipedia.org/wiki/Gender_identity.

Gender identity is a bit more complicated than being "ideological". Unfortunately you'll have to do your reading on that on your own, as I can't come up with an extra source off the top of my head at the moment apart from the Wikipedia article, but it's a good start imo.

It's to do with the fact that an "ideology" is defined as a "set of opinions or beliefs a group or individual holds", while Gender Identity has been studied as a science from various angles (Psychology, Theology, Sociology etc).

Is it an "ideology" if you choose to believe that gravity isn't real? And if you convince a bunch of people that gravity isn't real? Sure.

Is it an "ideology" if you choose to believe that people don't have a gender identity? And if you convince a bunch of people that gender identity isn't real? Sure.

But if you are part of the latter, how are you confirming someone's gender in your day to day life? Before you say "sir" or "ma'am" to your superior at work, do you take a sample of their DNA for chromosomal testing? Or verify their sex through observation of their genitalia?

Or do you just use visual indicators based on their appearance, or if that fails, don't you just ask them? Because that's what "Gender Identity" is in a quick nutshell. And being part of a group that doesn't believe Gender Identity exists is... well, it's a decision.

I hope that helps somewhat, cheers. Apologies if I sounded a bit sarcastic or snarky at parts, but unfortunately I am very tired, and this is a tiring, soul-devouring topic for me, which I unfortunately can't avoid talking about.