i think your points 1 2 3 are true in the sense that this is perceived, and that sometimes, the way things are being taught lead to this. but in my mathematics career, as well as in school, i never had the impression that these sentiments were actually the beliefs of teachers.

i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.

also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.

i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".

sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.