good game, but can't proceed without leveling crafting :<

[Reiterpallasch]

Why do something that I simply don't enjoy?

So don't do it? As people have already pointed out, the quest you're trying to do has nothing to do with the main story line.

And even if it was required...

and when I finally got it to level 10 after two days

Level 10 on any crafting profession takes no more than an hour at best, and maybe a few thousand gil.

I was shocked to find that I needed a gathering skill to get cotten balls

They're sold from the vendor at the weavers guild. The same vendor that the guildmaster tells you to talk to.