There are a lot of reasons Arle serves as sort of the ideal puzzle game ""rep"", partially because many puzzle games don't have this kind of identifiable character to tie itself to. Puyo Puyo is a distinctly character driven series, the cast is colorful and well defined. Each match of Puyo is simulating what's essentially a fight in-universe too - Puyo is a competitive puzzle game by design, high level play has even been compared to fighting games. So by default she's a pretty clear-cut puzzle game ambassador for, well, a fighting game. There was even a Puyo fighting game being plotted out at some point but abandoned because they felt it wasn't right with the brand these days... the source of many a frustrating argument back during Ultimate DLC speculation.
Moveset-wise Arle is also a mage, domething established as far back as the precursor Madou Monogatari dungeon crawler RPGs which feature much more direct combat and at times shocking violence. I think the Saturn entry in this series, which largely bridges the gap between the original Madou games and Puyo Puyo, does the best job demonstrating the kind of magic that Arle can use to supplement the more puzzle oriented mechanics.
So on top of being as influential as it is within its genre, I think it's also the quintessential "character-based" puzzle game and provides the easiest material to build a satisfying crop of Smash content.