When it comes to determining what counts as a chatzitzah for mikvah, we say that nails which are too long (the length that they would bother the person), then they have to be cut or they would constitute a chatzitza . Why don't we say the same when it comes to washing one's hands for bread, where chatzitzah is also an issue?
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The custom for women to trim their nails before going to the Mikva is recorded in Shulchan Arukh YD 198:18 but the reason given is related to avoiding dirt under the nail in the part of the nail which extends past the flesh. The Levush (ibid :18) there asks why this is not practiced by Netillat Yadayim. He gives two answers:
The Taz (ibid sk 19) gives another reason, namely, that since Netillat Yadayim is only rabbinic, we rely on the opinion of Rabbenu Tam that only certain kinds of very sticky/thick dirt create problems and we don't need to worry about people not noticing this under their nails.
The reason you stated for cutting nails is that of the Raavan as quoted in Shakh (ibid sk 25). However the Shakh ends up being lenient if the women did not fulfill this opinion of the Raavan and has already gone home from the Mikva and spent the night with her husband. It seems that certainly we can be lenient then in the case of the rabbinic Netillat Yadayim.
You should see the Beit Yosef YD 198:18 for more.