They climb on Link and honestly Link should react more, I mean it's a huge spider! It's as big as a chihuahua and it's crawling all over him. It makes more sense if it nips Link's hands, but these spiders like to hug and make Link fall.
In Twilight Princess they bite Link's hands, whereas in Skyward Sword, they climb on him. Can't imagine spiders that big would be light, nor would it be very fun to have those things scurrying along your body, so naturally you'd shake them off... I don't recall them biting Link in SS.
For me I'm not going to lie I killed them simply cause I have a phobia of Spiders and even in games I'm scared of them so you can image how fighting a giant spider boss goes for me.
*Me as Link spots Ghoma the spider Queen. Girly scream and starts running around the place avoiding being hit as I shot arrows, seeds and anything I can have Link throw.* The up side I learned the Deku Nuts stun Ghoma making her fall if she was on the celing. I also learned it will break the eggs she drops in OoT the N64 version.