There seems to be a mistaken assumption here that Republicans are one strand of ideology. They're hardly that. They are ranging from Trump's populist protectionism to the now nearly forgotten (inside the party) libertarian thinking.
Critics of Trump point out that he has also engaged in a domestic expansion of the state's role in the economy, or at least that he does that declartively, while at the same time some of his supporters (if not Trump himself) hammer on the idea that Trump would really support a free market, but that external forces (=China etc.) constrain him.
the Trump administration’s willingness to interfere with the free market is in many ways a natural consequence of the president’s economic nationalism. When the nation’s supposed interests are paramount, the state becomes the tool, and the market an obstacle to be overcome.
Traditional small-government conservatives, who have dominated the Republican Party in recent years, seek to make markets the arbiters and leave individuals and companies freedom to operate, with the government merely enforcing the rules of the road. Trump’s approach turns Republican orthodoxy on its head.
“A small-government conservative would, say, set up a process that avoids the government picking winners and losers. He [Trump] wants to pick the winners and losers. He wants to be the guy who says ‘I delivered those jobs’ or ‘I saved those jobs,’” said Phil Levy, a former George W. Bush economic advisor and now a senior fellow on the economy at the Chicago Council on Global Affairs. “It’s antithetical to the whole mindset of small-government conservatives.”
“There actually is little U.S. precedent for this approach,” said Adam Posen, the president of the Peterson Institute for International Economics. “The current Trump combination of very arbitrary state intervention, with a lot of company and sector-specific discretion by the president regarding tariffs and which countries to hit, is out there on its own.”
How has Trump’s statist approach to the economy managed to prevail with Republicans in charge of Congress for the first two years of his administration? On the one hand, Trump offered Republicans victories on many bread-and-butter issues they craved, including tax cuts and regulatory cutbacks across the board, in addition to a pair of conservative Supreme Court justices. Those Republican wins overshadowed the lurch away from market economics.
At the same time, Trump and others in the administration have often suggested that they are actually free traders and would prefer less of a state role in the economy. Trump himself sometimes talks of slashing tariffs to zero if other countries do, and surrogates continue to peddle the notion that Trump ultimately seeks free trade.
Those rhetorical bones tossed toward orthodox Republican economic views, Levy said, managed to keep opposition from within the party at bay for most of the past two years.
“People actually believed it, and that papered over the differences,” he said.
Posen sees Republicans, even after the era of Trump, returning to their roots—and no, not free trade or a hands-off approach to the economy.
“From the early 19th century until Reagan, the Republican Party was more protectionist than not,” he said. The current electoral shift toward more rural voters skeptical of globalization and free trade will likely reinforce that tendency for the state to displace the market.
“I think it is likely to last in the Republican Party for a while,” Posen said.
And Trump's apparent attempt to pressure the Fed to follow his will has some Nixonian reverberations:
And it's important that the Fed be independent both in practice and perception. We know what happens when it isn't. In the 1970s, President Nixon pressured Fed boss Arthur Burns to run an expansionary monetary policy in the run-up to the 1972 election. Nixon did this both through face-to-face conversations and hardball tactics such as leaks suggesting he was considering expanding the size of Fed or otherwise giving himself more control over monetary policy. And while we don't know for sure why Burns decided to run a loose monetary policy in an already inflationary environment, his actions "helped to trigger an extremely costly inflationary boom–bust cycle," concludes economist Burton Abrams in How Richard Nixon Pressured Arthur Burns: Evidence from the Nixon Tapes.
The US Libertarian Party claims it was formed in no small part due to Nixon's economic policies, in particular his announcement of wage and price controls.
To finally draw a parallel with Conservatism under PM May, one (sympathetic, albeit left-leaning) US commentator said May embraces
A working class conservatism [that] also means rejecting the libertarian temptation [...]
In the body of her speech, PM May sketches the public policies that flow from her vision of working-class conservatism: increased investment in affordable housing and infrastructure; a “new industrial strategy” that invests in industries of “strategic value to our economy”; good public schools for everyone; and social reform to reduce poverty among ethnic minorities and increase opportunities for everyone to attend college—including “white working class boys [who] are less likely to go to university than any other group in society.”
It also means making markets work for working people. Citing Edmund Burke, the Prime Minister reminded her fellow Conservatives that preserving something important means being prepared to reform it. That is why “where markets are dysfunctional, we must be prepared to intervene”—for example, where companies use opaque pricing structures to confuse consumers, where rural areas don’t have access to broadband, or where private capital does not give ordinary people a fair chance to buy homes. [...]
“Just listen to the way a lot of politicians and commentators talk about the public,” she declared. “They find your patriotism distasteful, your concerns about immigration parochial, your views about crime illiberal, your attachment to your job security inconvenient [...]”
The explanation, Mrs. May asserted, is class: “If you’re well off and comfortable, Britain is a different country and these concerns are not your concerns. It’s easy to dismiss them—easy to say that all you want from government is for it to get out of the way.” If the Conservatives are to be the party of ordinary working people, they must take populist concerns onboard without surrendering to the populist agenda—or to its least defensible sentiments.
“The central tenet of my belief,” she concluded, “is that there is more to life than individualism and self-interest. We form families, communities, towns, cities, counties and nations. We have a responsibility to one another. And I firmly believe that government has a responsibility too.”
No doubt the Prime Minister’s speech has the late Margaret Thatcher revolving briskly in her grave.
Of course one commentator's "working class conservatism" is another's populism, although there is perhaps a distinction of degree based on the level of discourse involved. Academic critics sometimes call this American brand of (working class) conservatism "Fox Populism", with reference to the Fox News channel.
Commentators sympathetic to Trump talk of
Trump’s Working Class, Conservative, Populist Realignment. [... and also claiming that] The Democratic Party is redefining itself, in part by relinquishing the working class contingent that was once the party’s bedrock constituency. [...]
Brownstein coined the term “coalition of the ascendant” to describe the voting blocs he saw as coalescing into the country’s dominant political force, including racial minorities, immigrants, millennials, and highly educated whites. And one more, which he identified in November 2012 in describing Barack Obama’s winning reelection coalition: “just enough blue-collar Midwestern whites to put the president over the top.” In other words, barely enough of those people voted for Obama to give him the battleground states of the Great Lakes region and hence to keep intact what Brownstein calls the “Blue Wall” of Democratic Electoral College hegemony.
But Hillary Clinton didn’t get “just enough” of those white voters. [...]
The Brownstein coalition stands for globalism, open borders, identity politics, free trade, cultural individualism, foreign policy interventionism, and gun control. Brownstein posits that this coalition’s growing force is driven by demographics—the decreasing “whiteness” of the U.S. population due to differential birthrates and the ongoing wave of immigration from non-Western nations. By 2012 this thesis was widely shared, including by Republicans such as Karl Rove and the authors of a solemn post-election analysis by the Republican National Committee. A Wall Street Journal headline over a Rove piece warned, “More White Votes Alone Won’t Save the GOP.” The RNC report declared, “The nation’s demographic changes add to the urgency of recognizing how precarious our position has become.”
For Republican mandarins this translated into an imperative to become pale versions of the Democratic modality, embracing globalism, adopting a soft attitude on immigration, doubling down on free trade, acquiescing in elements of identity politics, maintaining a stern foreign policy, accepting the industrial devastation wreaked by U.S. globalist policies—and reaching out beseechingly to all the elements of the Democrats’ coalition of the ascendant. This translated into a view that, as Zito and Todd described it, “the only possible winning future Republican coalition must, by dint of math, become less white, less old, less rural, and more educated.”
And then along came Trump, the candidate of infrastructure spending, immigration curtailment, protection of entitlements, a ruthless assault on the Islamic State, selective curbs on free trade, Second Amendment gun rights, and foreign policy restraint. He not only laid waste to the Republicans’ “me too” drive to chip away at the coalition of the ascendant, but he did so with a raw contempt mixed with a scabrous mode of expression that was offensive to many but struck others as demonstrating a resolve to shake up a political establishment that had become ossified and oppressive.
In the process he demonstrated that Brownstein’s ascendant coalition concept was at least premature and possibly flawed.
And in the UK, depending how one defines "working class", either the Conservatives or Labour can claim the crown of representing them.
In summary "Republicans [...] favour free markets" is a rather amorphous description. The winning Republicans (or UK Conservatives) don't insist on it as much as the (mostly losing right now) libertarians in the same parties do.
And if you take the view that markets can be dysfunctional, it's no biggie to then assert that this or that issue (that bothers you) is a result of non-free forces.
Psychologically speaking, assignment of blame to impersonal forces is more associated with (mental) depression, while assignment of blame to someone else is more associated with anger. Which of these two states is more likely to energize voters is a bit of a no-brainer.