Why do most Republicans claim to value X
Short answer: Because right-wing propaganda convinced them to value it.
Remember there are hundreds of millions if not billions spent each year, powered now by big data psychometrics that make old fashioned marketing look like child's-play.
Many wealthy/business types vote Republican for the tax cuts and the de-regulation, and keeping their target markets less powerful.
The rank and file Republicans don't really know what any of this is *really* about. They language they have been fed to think about it is like this:
liberal/Dem - socialism, nanny state, victims, takers
conservative/Republican - freedom, independence, liberty, free markets, America **** YEAH!
Given such an absurd false choice, its no wonder they oppose themselves. Hard to discuss wage gap, income disparity, etc., when that's the language they have to describe it.
Income disparity is most easily addressed with a combination of these which are already in use.
- appropriately regressive taxation
- business regulations, including treating large natural resources (think energy) in some proportion as public wealthy/resource
- organized labor w/more labor protections, etc.
Both are traditionally opposed by today's Republican party. We just had enormous de-regulation and tax cuts for the wealthy and corporations. This gives a short-term boost to the economy, which turns out the same disparity going mostly to the wealthy, and a fraction to workers, but the federal debt increases, which is ultimately paid by everyone..but today less by the wealthy/corporations. This creates a worsening long-term wealth situation by the average worker.
Big energy was de-regulated, EPA was gutted, that will damage all our land and water and air more...we all bear that cost, but big corporations get the lions share of the profit from it.
That's my take anyway.