One thing that is particularly arbitrary is that C is named "C" rather than "A", which would make far more sense (given that it is the root note of the only major scale that has no sharps or flats).
It would certainly make more sense for people learning music. The C major scale is universally agreed to be the simplest, most basic scale, so wouldn't it make more sense for it to be ABCDEFG, rather than CDEFGAB?
The question is, "Why are the notes labeled the way they are, instead of some different labeling (which would change their relationships between each other)?" This has nothing at all to do with "calling a clearly minor scale major". Let's suppose we relabel the notes, so that C is now named "A", D is now named "B", E is now named "C", et cetera. In this alternate universe, Ab is enharmonically equivalent to G, and the scale "ABCDEFGA" is a major scale, and "FGABCDEF" is the corresponding minor scale.
Part of the question is, "Why did we name the notes the way we do, instead of that other way?" That's actually an interesting question, once you get down to it.
Another part of the question is, "Isn't our system kind of arbitrary, and doesn't this alternative universe make more sense?" The answer is "No, the alternative universe isn't inherently better or less arbitrary, for the various reasons we talked about in this thread."
The main two reasons why we wouldn't prefer one universe over the other are because (1) both the Ionian and the Aeolian modes are important in western music, and it's difficult to claim that one is more important than the other, and (2) you have to learn a bunch of other scales anyway, and if you have a hard time with CDEFGABC then you're never going to make it through other basic scales like G major or F major.
(Bug... the "flat" symbol seems to be getting stripped out of my post... so forgive me for using "b" instead U+266D)
Maybe, yes, you still learn the C scale first, but you'll learn the A scale the next week. In the grand scheme of things if you are having problems remembering the names of notes then moving the keys over two steps is not going to make the necessary difference.
No, it's not. There is no such thing as a "simplest, most basic scale." That depends entirely on your arbitrary choice of notation and conventions.
So it's legitimate to ask, if we're going to letter our notes based on the notes of one key, why not pick the key that sounds the most basic and fundamental? (The answer is that at one time folks heard music in a different way and Ionian didn't have the priority that it does now; see my other comment [1]). It also probably doesn't hurt that Aeolian (natural minor) is arguably the second most natural and "default" sounding mode, so it's not nearly as weird as if we decided to letter the notes off of Locrian or something.