That analysis ignores who the Third Party candidate is: A popular leftist Third Party candidate would indeed take votes from the Democrat and help elect the Republican. Likewise, a popular right wing candidate would take votes from the Republican and help elect the Democrat. In general, a Third Party candidate can only help elect the mainstream candidate the least like them. This is why people say Third Party runs can only hurt their cause.

It would be completely different if we had a parliamentary form of government, where a Third Party could form a coalition with the mainstream Party most like them, But we don't.