Like I live in NYC, pretty much every conservative is a registered Democrat so they can vote in the primaries. The Republican party will usually get more Democrats votes here then any third party in presidential elections
That doesn't mean a third party can't attract voters who would go for the Dem
Clinton was the last Democratic candidate to have success in a lot of the South, where the party had been losing vote share among registered Democrats since the Civil Rights Act
But I don't understand why you think Nader, a third party candidate to the left of Gore, did not mostly attract people who would have voted for Gore over Bush if he wasn't running
...because Gore lost Democratic voters to Bush. a very large number of them, much more than Nader "took." if Gore hadnt lost them, he would have defeated Bush easily. That is on Gore and the Democrats, not Nader.
if 0 Dems voted for Bush, you could pin the lost on Nader. but they didnt.