Rubio ran for the Republican nomination in 2015-6. He got lots of media coverage and seemed to be well financed, but didn't get many votes. Those that he got weren't mainly from the working class, but from "establishment Republicans." That is, when working class voters were offered a choice between the leading "pro-worker conservative" and Donald Trump, they unhesitatingly went for Donald Trump. I don't find this hard to understand--if you read the elite media, you knew that Rubio was supposed to be a reform conservative, but in the debates he was just one more guy talking about how conservative he was and how Hillary Clinton would do irreversible harm to the America we knew and loved.
What would a pro-worker conservatism be like? Douthat links to a piece by Pete Spiliakos, which says that the problem with the recent tax bill is that its benefits are skewed towards high earners, with too little going to the middle and working classes--in other words, exactly what Democrats are saying. A reasonable short definition of the difference between left and right on economic issues is that the left is in favor of using the power of the state to help people with low and moderate incomes at the expense of people with high incomes, and that the right opposes that. So a pro-worker conservatism would have to involve some move to the left, because being pro-worker is a basic principle of the left. Of course, it could involve more than that--for example, expanding the Earned Income Tax Credit but cutting back on the minimum wage, or reducing occupational licensing--but it can't avoid it entirely.
However, as I have observed before, both politicians and intellectuals in the Republican party seem to be consumed by the desire to prove how conservative they are, and how strongly they oppose progressivism and all of its works. With politicians, this means that if someone compromises, he or she is vulnerable to being pushed out by someone who promises to take an even harder line. With intellectuals, it means that even those who want reform assume that any reform has to come from the right and therefore convince themselves that people like Rubio, Lee, and Cotton might be the answer. The result is that both establishment and reform conservatism are ineffectual.