I've tutored many college students (of various ages) in basic algebra, almost all of whom were convinced they couldn't ever possibly learn it, and only one of whom didn't end up getting an A in their course. I'm pretty aware of the struggles some students can have with it and none of them really had trouble memorizing the handful of symbols that are actually used at that level. Some struggled with the concept of a variable, but many just struggled with understanding the relationship between the actual concepts and the manipulation of symbols. Off the top of my head, equations being balanced can be a difficult one, but certainly not the only one.

As for younger students, I have much less experience, but some; and it's interesting you mentioned less than and greater than; since I actually remember learning those symbols. We learned that the "alligator" always eats the "bigger" number. It's not surprising there are countless ways of learning it, including song. That's true of almost any abstract concept. The idea is to link a metaphor the person understands to the abstract concept. Not every metaphor will work for every person; and this is true of all abstract concepts, not just math symbols.

Of course, it's not actually true that the alligator is eating the "bigger" number, and it actually demonstrates why we need the symbols. ">" and "<" actually refer to "greater than" or "less than" which we much later learned is a way of saying "which number is further right on the number line"; which, of course, requires the abstract concept of the number line and accepting the more-or-less arbitrary decision of a left-to-right number line. "Bigger" means "has a greater distance from zero on the number line in either direction" which we'd represent as a comparison of absolute values. I don't recall when I learned about absolute values, but it was definitely years after learning about < and >. Using the proper symbols lets us be explicit, concise, and precise and avoid issues like using English synonyms (bigger, greater) or whatever pitfalls exist in other languages.

The choice of symbols < and > are, of course, largely arbitrary other than the symmetry between them. (We could have, for instance, always put the greater number underneath the smaller number so the structure is more stable in an imaginary gravity; but that, too, would be arbitrary.) But so is the letter S, or the number 9. They're all arbitrary symbols that have particular meanings in particular languages. "9" is interesting, because it's a number, versus the Roman numeral system. The Roman numeral system could arguably be called non-arbitrary. "I" clearly represents a single thing, "II", two things, etc. That works until you get up to "IV". What? "IV"? So if a lesser value is in front of a greater value, you subtract it? And how does "V" represent five anyway? It's arbitrary! But the Romans found it much more useful to be able to write VII + VI = XIII rather than IIIIIII + IIIIII = IIIIIIIIIIIII, which can pretty quickly get unruly. Turns out, memorizing digits 0-9 makes it (and more complex math) even easier: 7 + 6 = 13; which is why the entire world uses numbers instead of numerals.

(We could also have a side-discussion on why base 10 and not something like base 12, binary, a mixed radix system that uses the prime numbers or the sexagesimal system used by the Sumerians. The answer is basically that its mostly arbitrary, simpler than some systems, and we have ten fingers/thumbs.)

> Why make "change" Δ

> Why make "square root of -1" i

Largely historical reasons, expediency, and lack of better alternatives. Why represent the sound "ssss" with the symbol "s"?

You could swap i for √-1 and people would understand you, but you'd very quickly wish there was a shorthand that you could use to represent this rather special value.

> In how many classes do we see rote memorization of the quadratic formula, with no context around why you should even bother to learn it? (I've seen quite a few).

You won't see me objecting to this; but this is not an issue with mathematics. It is an issue with teaching mathematics and is part of the "lament" I linked to above. This is quite a different issue than the issue of symbols. The symbols, while arbitrary and arcane, actually make the mathematics more manageable and precise. Saying that "facilitates laziness" is like saying a clothes washer facilitate laziness since it removes the need to manually provide friction and agitation. It's true, in a sense, but I'll keep my washer.

Mathematics is taught very poorly in many places; but making it hopelessly complex and less precise by removing the symbols of the language is not going to help. I learned algebra, officially, my freshman year of high school. Yet there are many high school graduates who come out of high school not even having a rudimentary understanding of algebra (and, actually, even basic mathematics - tutored a few of those as well). Many of them learn it in college, so they're obviously capable of learning it. Those high schools failed those students. Many university professors equally fail their students.

But blaming this on the symbols of the language is too far of a stretch for me. Blame the teachers.

Edit: Fixed display of symbols.