Thank you!
Well, even with 2d4+1 and 2d6+2 damage, stationary bowmen outdamage crossbowmen against unarmoured foes (especially if they have weapon mastery!). It's the guaranteed minimum damage of 3 and 4, respectively, that would give crossbows a distinct and measurable advantage (and the fact that heavy crossbows, albeit only every other round, have a chance to deal a significant amount, making them more visceral).
I agree, though, that damage output isn't the only thing that defines a weapon: there's also cost, weight, style, and other hardly measurable advantages (such as that axes can be used to hack open a wooden crate or door, maces are easier to smash things with, and crossbows are ideal, as you pointed out, for lying on a roof with an arrow cocked, ready to assassinate).
I have a sweet spot for B/X D&D in my heart, where everyone attacks only once per round and only d4-d6-d8 damage dice exist (at least for players) - a very intuitive system. AS&SH, on the other hand, has more options, more varied attack rates and damage dice, and I felt the need to investigate whether certain options worked as I anticipated them to work. I have no intention to "balance" things; rather, I had a vague feeling crossbows weren't actually that great (and maybe that a bunch of crossbowmen shooting at my players wouldn't be as threatening as I hoped?), and I needed data to support that.