Rip, was Byrd the leader of his party? Is Lott resigning from the Senate? Is there even a national outcry from him to retire from the Senate in the media or otherwise? I'm sure this will be an issue in his state, but then that's between him and his electorate, the same with Byrd. Byrds remarks were hardly overlooked, it's just that even though he is a powerful Senator, there are a lot of powerful Senators. There are a lot of Senators and Congressmen, period. However, there are only a few politicians who represent their parties at the national level and Lott is one of those.
As for Clinton, I'm sure the Democrats would have hung him out to dry if they felt he was more of a liabilty to the party (and their own chances at reelection). But the public wasn't all that outraged. Much of the public saw both the republican attack on his sexual exploits and his comical defense as politics as usual (as opposed to, say treason), and somewhat disagreeable as a whole. And, there was no real political opportunity to be readily gained fighting Clinton within the party. Had that been different, then I'm sure the Democrats would have been just as nobel.
Charon