I probably won't convince you, but let me try to convey one more time that it functions as one word, thusly:

"My son is in /'haisku:l/."

"3800 feet up in the mountains? That sure is a /'ha:i/ /'sku:l/!"

In some languages, the word for word translation for high school refers to something else. E.g. the Slovak phrase "vysoká škola" literally means "high school" if taken word for word, but actually it refers to some post secondary higher education. This corresponds corresponds to the German word Hoschschule which looks like a cognate for "high school", but meaning-wise isn't. High school is called "stredná škola" which literally means "middle school"! (And the university-prep variety of such a school is also called "gymnázium", the cognate of which denotes a sports or exercise hall in English.)

So the "high" in "high school" is simply not directly informative; you only know what kind of school that refers to from your cultural knowledge and context.

I find it strange that you listed high school as being distinct from high class, when both have the same non-literal meaning of "high".

Maybe it's a cultural thing, but I pronounce both high class and high school without a pause, and with emphasis on both the high and the class / school.

Must be a highclass phenomenon.

that's right, it describes the state of mind of half of its visitors ;)