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 I think the South's history makes party registration in the second half of the 20th century less significant than you think it does. But I don't think I'll convince you and I don't think you'll convince me so I'm going to just say goodbye now