I could use alpha - a as in add.

I could use sigma - s as in subtract.

I could use iota - I as in integrate.

Of course it's arbitrary.

You claimed the symbols were arbitrary, they are (often) not. What you demonstrate here is not that the symbols are arbitrary, but that the concept (sound as abbreviation) maps to more than one potential meaning and the choice of which one is standard or convention is arbitrary.

That is correct, but at this point it is standard to see sigma as summation (or in some other contexts as set or space, but it's also clear in those contexts which meaning is appropriate, certainly an algebra student seeing sigma notation will not also be seeing sigma used for "set" or "space" in that same course).

You're conflating choice with canon/convention. The choice was made arbitrarily by someone. There was no convention at the time the arbitrary choice was made. The choice to use alliteration was as arbitrary as the choice of which Greek letters to use. There are other equally arbitrary possibilities.

Also don't confuse arbitrary with random. Having a reason doesn't mean a choice isn't arbitrary.

Bigger picture, maybe I'm misinterpreting you, but I feel like the logical conclusion of your argument is that nothing can change now, because canon. And yet basic math notations have changed throughout history.

It would take a lot of effort to change the summation symbol from sigma to something else. But if something else made learning math faster, the canon is exactly the part that needs to go, not an excuse to stop improving.