For example, has any Republican ever seriously researched the question of what level of salary for officeholders is best for attracting better Republican candidates and officeholders?
Most discussions of pay for politicians in America seem to be moralistic/idealistic rather than pragmatically partisan. I can't recall anybody saying we should pay state assemblymen more/less because that would help my party beat the other party. But, that is an interesting question.
By way of analogy, about a century ago in Britain, the question of whether the government should start paying a salary to Members of Parliament was a major partisan issue. The fledgling Labour Party desperately needed to have the law changed to have salaries paid to MPs because it did not attract many rich supporters who could live off their "private incomes." Eventually, the Labour Party got the law changed and was able to recruit talented candidates to be professional politicians, dooming the Liberal Party. In other words, this is serious stuff.
In the U.S. today, the income gaps between Republicans and Democrats in a particular locale aren't as large as between the the three British parties a century ago.
Still, Republicans, I suspect, would be better off if governments offered either very low or very high pay to entry level offices, such as state legislators. Too often pay gets stuck in-between where it's very attractive to the typical potential Democratic politician (e.g., the chairperson of the Diversity Sensitivity Nook at a community college or the press spokesperson for a Save-the-Sea Otters organization, and would be making five figures) but not attractive to a potential high-quality Republican politician (who would typically be running a successful business unit and making six figures — or is married to somebody making high six figures and doesn't need the salary).
For example, a quality GOP candidate might be, say, a CPA who has built a decent-sized accounting business and now employs other CPAs to crunch the numbers for him while he largely functions as a rainmaker through his multitudinous business and charitable contacts in the community — somebody good with both numbers and people. (Maybe that's not a good candidate — that's what research is supposed to figure out).
For the Diversity Sensitivity Nookmeister and other potential Democratic candidates, a state assemblyperson's salary might be a major step up in life, while for the successful Republican CPA, it would require a crushing sacrifice of his family's well-being to take the pay cut.
So, the GOP, being the business party, ends up with a lot of candidates who weren't very successful in business, and probably won't be very successful in running for office or in office either.
Perhaps instead of moderate pay for both lower and upper house members in the state legislature, the GOP would thrive best with low pay for assemblymen and high pay for state senators.
I don't know. I'm just speculating. But that's what research should find out.