Look, the nails have to come off somehow. I don’t think biting is any more disgusting than using a clippers. Just don’t do it in public, properly dispose of the nails, and wash your hands afterwards.
My number one gross-out complaint is about people who put dirty tissues on tables in shul or leave them in shenders in yeshivos. They should go straight to the pocket. Who wants to put his hands on a wet table where somebody just blew his nose. In addition, one should wash his hands after blowing, or getting them sticky with food.
The gemara says it is very wrong to gross someone out and gives two examples. One who spits or kills a lice in the presence of another (Chagiga, I believe).