This is a very good question, which has been fully addressed in [בני בנים ג:ו 
ד"ה עוד קשה לי][1]  by Rav Yehudah Herzl Henkin, who concludes that nails do not actually pose a problem of *tumah*.

[1]: https://www.hebrewbooks.org/pdfpager.aspx?req=20024&st=&pgnum=26&hilite=