I think you are getting good advice here.
I’ll add a similar-but-slightly-nuanced take on it.
Reaching out to a teacher/club advisor makes sense if you have a true objective/question, and if you are getting more information that helps you determine your own fit at that school. (And of course – do NOT reach out to ask about anything that is on the website already.) If you, as a result of the connection, now know more about a program, and see some specific ways that it is actually the perfect fit for you, then that fuels your own fire for truly believing that you two are meant for each other. And THAT can absolutely help you get in because you will be better at explaining how you are a fit. If you hear nothing else hear this: THIS IS HOW YOU GET INTO A SCHOOL – They see you super clearly succeeding and contributing on campus.
BTW – this is also true for doing well once you are in school. I am a huge fan of connecting in academia – office hours, etc. Not because a professor will give you a better grade simply because you show up to office hours. But because you gather information in direct connection that then prepares you, the student, better for tests, papers, class participation.
Same thing here.
The play is NOT “reach out to the teacher so they contact admissions.” It’s far more organic (subtle?): Dig in to determine if you really are a deep fit. And that will help you make the case for admissions that you belong there. (And yes I would also go through AO to do that kind of outreach. For two reasons: 1. more respectful and 2. It also subtly lets your AO know that you are deeply interested.)
Finally, on this point, if you ask any time of anyone, YOU take the responsibility for having a purpose. And if it’s deep-down really just a thinly veiled hope to get a leg up in admissions, this will not be well-received. If you don’t have anything, truly, to ask or discuss, then don’t reach out. Don’t put the burden on THEM to come up with a purpose for your meeting. (This looks usually like this: Student says something vague like “tell me about your robotics club.” That is the language of putting the burden on the other person. Do not do that. But saying: “I saw that your robotics club went to the World Series of Robotics in Romania last year, and I wondered if that meant you followed Eastern European circuit protocols because I’m interested in international electronics, and I don’t think I’ve ever seen a secondary school teach both Western and Eastern theory.” NOW, you have a conversation.). (Albeit not a good one since I was just making something up, of course.)
FWIW – my DD2 last year reached out to zero academic folks and just a couple of coaches (before she realized she was absolutely not in varsity-recruited level of her sport so she stopped coach-contacting), and it did not seem to hurt her admissions at all.