Why Not Run as a Republican?
As I was going through social media posts recently, I read a question that was posed to a former Libertarian candidate for a state office: Why not run on the Republican ticket, like Sanders did on the Democratic ticket? I answered the question in her comments, and since the same question had once been asked of me, I figured I should post the same answer here (but this one is a bit more expanded).
So, my answer to the question was, in a word, "ethics" defined as "moral principles that govern a person's behavior".
While it may be true that some Libertarian ideals are similar enough to those of the Republicans to allow for the possibility, there are simply too many Republican positions that small-letter libertarians (those who believe in the principles of liberty, but who are not members of the political party) find so repulsive that any truly libertarian candidate running as a Republican would probably be defeated in the primaries.
Then, there's the stated Libertarian intent to dismantle the two-party system. I will admit that an internal destruction of either party would effectively achieve that goal, but that would involve "spy-like" tactics more effectively practiced in the literary world than in the political world. I will grant you that some future candidate may be able to bridge that transition, but most of the candidates that I have the honor of knowing cannot (nor would they if they could). As a candidate myself, I know that I cannot. The "wolf in sheep's clothing" persona that I would need to adapt is one that I also know that I would never be able to effectively convey.
"The price of what we're winning is the same as what we've lost."
As for Sanders running on the Democratic ticket, Sanders is a declared Independent despite being a Democratic Socialist ideologue. As a former member of the House and as a Senator, he has often caucused with the Democrats, so they tolerate him because he brings them money and votes. But he is not a Democratic Party member so, as far as they are concerned, he can never be selected as their nominated candidate (which we saw in 2016). I suspect that such would be the same for a declared Libertarian trying to run on a Republican Party ticket.
The Libertarian Party slogan is "The Party of Principle". If I'm going stand on the principles in which I believe, why would I start by associating myself with a political party in whose principles I don't believe?
Exploring a Tangent
Some may view that idea of dismantling the two-party system as simply a "temper tantrum" by the Libertarian Party (and other third-parties) mostly because Libertarians have a long track record of being defeated against Old Party candidates, but I look at differently (I'm autistic, so I do that a lot).
I see the ideals of each of the Old Parties as both wildly divided to the point of extremism, and solidly united behind a commonly-held idea. The ideological division is obvious, but the ideological unity is not, and that idea is "dominion". Ideologues holding beliefs from either party, even if they are not politicos, are united behind the idea that only these two parties (and therefore, their opinions) are valid; all others are suspect and must be removed from popular consideration by any means available (which we also saw in 2016).
"All is for the best. Believe in what we're told. Blind men in the marketplace buying what we're sold."
"Temper tantrum"? Perhaps. But while the Old Parties loudly laud their continued electoral successes, they are also anxiously whispering about how those successes are achieved as a result of repeatedly lower ballot counts. Greater numbers of eligible voters are simply not voting, and they cannot seem to figure out why even though they have been repeatedly told the answer: those voters simply don't like either of their candidates. Oddly enough, I still think that officials in both of the Old Parties cannot grasp that concept, so I don't see it changing any time soon.