(1) Certainly, there are Republicans who favor a relatively high minimum wage. They are simply Republicans whose understanding of basic economic principles is, shall we say, stunted.
(2) I'm sure you could also find a Republican elected official or two who has sponsored or voted for an increase to the prevailing minimum wage. This is a manifestation of political pragmatism.
(3) Just as there is no debate among PhD biologists about the 'Theory of Evolution,' there is no debate among economists about the minimum wage. It is universally understood among economists that the minimum wage itself is a colossally stupid idea that can only be viewed from a standpoint of minimizing its harm.
(4) The American population is comprised of all sorts of people, and there is a large group of them - many of whom are registered voters - who lack the basic intelligence or experience to understand why the minimum wage is a bad idea. Democrats traditionally cater to The Stupid, and it is in this spirit that they perpetually favor "increases to the minimum wage." But it is a sham. Do recall that there was time shortly after Barry was elected President that the Democrat party controlled both houses of Congress and the White House. And yet, when they had the power to do so, they didn't even address the minimum wage. It is a puzzlement, is it not? Unless you understand that even Democrats in Congress know what a stupid idea a "high" minimum wage is, and they understand that it would put millions of poor and/or minority people OUT OF WORK.
(5) One must understand that in every economic locale there is a LEGAL minimum wage (usually the Federal minimum wage), and there is an ECONOMIC minimum wage. In New York City, for example, no one will work for less than $12/hour, so the legal MW of $7.35 (or whatever it is) is irrelevant. The NYC government could raise its local MW to $12/hr and that legislation WOULD HAVE NO EFFECT ON EMPLOYMENT in NYC. OTOH, in most of the rest of our Great Country, the economic MW is pretty close to the legal MW, so a SIGNIFICANT increase in the legal MW would have ripple effects. Employers would insist that employees work more efficiently so that hours and/or people could be cut; they would look to automation for the purpose of minimizing headcount (e.g., self-ordering kiosks at fast food joints); they would look to "black market" employees (illegals working "under the table"). In fact, people at the bottom of the ladder - high school kids working part time, dropouts, and young HS grads with no education or work experience, would suffer greatly, as they do now in minority neighborhoods.
The Minimum Wage debate is a kabuki dance performed for economic idiots who happen to hold voter registration cards. Neither Democrats nor Republicans favor a "high" minimum wage because they both know that it is a bad idea. But the Democrats pretend to favor it in order to fool gullible voters into thinking they are "compassionate" and "caring."