> If there's one thing both parties agree with, it's that you can't ever vote for a third party

Actually, both major parties (not always at the same time) have a long track record of working very hard to promote voting for third-party candidates, doing things like funneling funds covertly (or simply nudging donors) to fund their efforts, assigning party activists to support third-party efforts, etc.

Of course, they exclusively do this for third parties whose appeal is, or is expected to be, mainly to people whos preference, if choices were limited to the major parties, would be for the other major party.

Because it's not just rhetoric, as long as the electoral system isn't reformed to change this, getting people to vote for a minor party instead of your opponent like demoralizing them and getting them to stay home, or disenfranchising them (two other things the major parties have been known to try to do to populations likely to vote for their opponents otherwise) is a lot easier and exactly half as useful, per voter, as getting them to switch to you from the other major party.

That only works if the message of the third party is more appealing to those voters. And so the major party also pays attention to which third party messages from those who would support them are getting through and changes.

It is also helped because many of the people who are insiders in the major party are secretly voting for the third party when the majority of primary voters (who are rarely well informed) force someone they don't like on the party. They can't do anything this time, but they can send a message to each other where they failed.