It's not just about beauty, but experience. Imagine you have a spreadsheet with infinite rows where each row contributes to the Higgs mass. It also turns out that by and large each row contributes less than the previous. You can't calculate all of them, but calculating the bigger ones you can do pretty well. Now imagine someone plops in infinity into one of the rows, sure the rest could all just happen to contribute a small amount and conspire to add up and cancel that out (because the mass isn't infinite), but it just doesn't seem like the world we live in. However, if you add another column to your spreadsheet for super symmetry and suddenly that column naturally cancels out the discrepancies from the first column. It's a simplistic example, but hopefully gives a better idea why Physicists were hopefull for some kind of new physics.
This is called the hierarchy problem. https://en.wikipedia.org/wiki/Hierarchy_problem#The_Higgs_ma...