This is a very good question, which has been fully addressed in בני בנים ג:ו  
ד"ה עוד קשה לי https://www.hebrewbooks.org/pdfpager.aspx?req=20024&st=&pgnum=26&hilite= by Rav Yehudah Herzl Henkin, who concludes that nails do not actually pose a problem of *tumah*.