Counterpoint: anybody not cheering for Fox News' talking points sufficiently enthusiastically is declared "socialist" and all but suggested as target for public lynching (or in some cases actually the target of a lynch mob storming the capitol) and the masses have been drilled into going berserk at hearing the right keywords, regardless of what happens in reality.
Calling an attempt to break this information monopoly over half the nation an uphill battle is the epitome of understatement.
Honestly, Fox News is going to call anyone against them a socialist. That's just what they're going to do. There are even some Republican policies which can be construed as socialist, such as public spending on schools, even private schools.
The average voter really doesn't care about if something is socialist or not. They clearly don't care if something is fascist. They only care about the perception of how it will affect their life.
If you tell people they're not going to have to pay for healthcare ever again, or that cannabis will be legal, or that you will fix the massive wealth gap, they will vote for you. Nominal Republicans would even vote for you. The downside of using the word socialism like that, is that Fox News that has removed the meaning of socialism from their base, meaning that actual socialism will not seem like such a big deal.
Being called socialist isn't even the issue. My point is that a news monolith has spent decades radicalising half the country and at this point there are enough cultists that will mercilessly tear down anybody Fox&Co paint a target on. Until there is a solid defense against such a hate machine, no party will flourish on its merits as long as Fox-drilled stooges will only be fed a caricature of any opposition Fox designates as the enemy. Even your perfect ideal candidate would lose the election. The boogeyman Fox would paint them to be would be so repulsive and half the country would never even hear their ideas anyway, except what skewed perspective Fox would blast the viewers with 24/7.
My point is that they're construction is inherently self-destructive. They have made their base extremely susceptible to populism.
If a populist candidate from the other direction arises, a large portion of the Fox News base will support them. You mark my words. Just think about how quickly they turn against Fox News when they don't do what they want