Multiplication order in current mathematics standards should happen the other way around when it’s in a non-commutative algebra.

The good thing about multiplication being commutative and associative is that you can think about it either way (e.g. 3x2 can be thought of as "add two three times). The "benefit" of carrying this idea to higher-order operations is that they become left-associative (meaning they can be evaluated from left to right), which is slightly more intuitive. For instance in lambda calculus, a sequence of church numerals n~1~ n~2~ ... n~K~ mean n~K~ ^ n~K-1~ ^ ... ^ n~1~ in traditional notation.

For example, we can’t write 2ω for the next transfinite ordinal because 2ω is just ω again on account of transfinite and backwards multiplication weirdness, and we have to write ω·2 or ω×2 instead like we’re back at primary school.

I'd say the deeper issue with ordinal arithmetic is that Knuth's up-arrow notation with its recursive definition becomes useless to define ordinals bigger than ε~0~, because something like ω^(ω^^ω) = ω^ε0^ = ε~0~. I don't understand the exact notion deeply yet, but I suspect there's some guilt in the fact that hyperoperations are fundamentally right-associative.