I remember watching an interview of Richard Feynman where he mentioned that when he was a kid, he figured out his own way (invention) of doing logarithms and he made up his own notation and names. Then, when he tried to explain it to other people he realized it's important to have names for things that everyone agrees on. This is a good reason why we have to teach the "arbitrary". As much as I'd wish the nomenclature be changed, we still have to communicate to each other and we have to teach it. This is the same case with mathematics conventions, and defining certain concepts in some way, for example negative numbers being to the left on the number line.
I agree with the idea of not teaching "necessary" aspects as "arbitrary". I do think that a lot of teachers do this however. I made it my goal to guide the students to derive everything, and not just give out formulas. I think it's possible with most mathematical concepts in high school to do this, and most derivations don't require much sophistication.
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