I see. They're two sides of the same coin in the sense that they're both part of the two party system. My original comment is overstatement and maybe melodramatic. Sorry for that, and for the accusation, and thanks for the civil, thoughtful response.
The other comments do a good job of explaining why an alternative party hasn't emerged, better than I probably can, so I'll skip that part. To some extent I do wish (and I gather you share the wish) that the US political system worked more like Europe's multi-party parliamentary democracies, relying on shifting, unstable coalitions rather than monolithic, monopolistic party machinery. On the other hand, I think it was Europe's parliamentary system that preceded, and produced, the Third Reich and other fascist regimes in the early 20th century.
In fact, I wonder whether two-party systems, like the US, on average produce worse outcomes than multi-party parliamentary systems. I'm not sure they do. But I also don't know enough about politics, political theory, or modern history to answer the question myself. I'm not even sure which other political systems are, or were, two party.
Edit:
On second thought, I'm not sure I agree at all with the other comments that explain why alternative parties haven't emerged. The comments all take for granted that there's a desire for an alternative but also that the alternative wouldn't be viable. I'm not sure there is such a desire. Most polls show that GOP voters approve of the party, if I'm not mistaken. So the answer to your question may be a lot grimmer than the one already offered here: there's no third party, very simply, because the overwhelming majority of GOP voters really do want a fascist regime.
On that note, this article is worth reading: https://www.vox.com/2016/3/1/11127424/trump-authoritarianism . It makes a great case against the materialist explanation of authoritarianism's rise in the US (i.e. the claim that Trump voters are angry about their worsening prospects, declining fortunes, and deteriorating communities, often related to the opioid epidemic). Instead, it explains support for authoritarianism as a result of disposition and psychology. That rings true to me. American authoritarians really do care about the things they say they care about -- above all, "woke" politics, transgender people in bathrooms, immigrants, Muslims, people of color receiving preferential (i.e. fair) treatment. They really are fighting a cultural and religious war, not struggling against unfair, challenging economic conditions. They really are just hateful.
They hate secular progressives and want to shut them out of the political process -- and want to brainwash their children.
They hate LGBTQ people and really do want to push them back into the closet -- and ideally wipe them out.
They hate people of color, or those who seek equality, and see nothing wrong with the disadvantages people of color face. Out of one corner of their mouths they'll scream about tradition, their pride in "their" country, and how hard their parents and their parents' parents worked -- and then out of the other corner of their mouths they'll reject that America's history of slavery, Jim Crow, racism, and the like have any continuing relevance or consequences for people of color.
They hate immigrants (or rather brown immigrants) and really do want to close the borders.
They're driven, in short, by a primitive xenophobia triggered by anything different from themselves -- which is why Fake Tan President is their God emperor. He's one of them.