Republicans have been hurt by Plurality voting many times.
A long run of Republican dominance was broken in 1912 when Democrat Woodrow Wilson was elected due to a vote-split between old-style Republican William Howard Taft and progressive-style Republican (running under the "Bull Moose" flag) Theodore Roosevelt.
The totals were
Did this hurt the USA and the world? Probably. Wilson in his second term was a mental basket case due to a stroke. Like usual the Democrats conducted an immense cover-up to conceal his health problems (just like with F.D.Roosevelt and J.F.Kennedy later), causing (it is thought) his unelected wife to basically be in charge. Wilson devoted the last of his mental and physical strength to trying to push for the "League of Nations," a harebrained and naive idea that the USA rightfully rejected. Roosevelt if elected would not have been a mental case, probably would have been better prepared to handle the challenges of World War I, and would probably have done more "trustbusting." Indeed many regard TR as one of the USA's best presidents, which is why he is on Mount Rushmore.
The Republicans were so shell shocked by this experience that, next election in 1920, they became determined to unify behind the most mediocre possible milquetoast principle-free nonentity they could possibly find: Warren G. Harding. (In the words of one of his close associates, Harding's only qualification for office was that "He looked like a president.") This eliminated all possibility of internecine rivalry, since Harding was far too boring to actually take a stance on any issue and had a long track record of doing nothing (he stayed away from US Senate sessions discussing anything controversial, like Prohibition or Women's suffrage – giving him the probably the single worst attendance record in the Senate during his years as Ohio senator) while being very pleasant and trusting – so nobody could find a way to object to him. And of course the USA was sick and tired of Wilson and his League. So sure enough, Harding was elected by one of the biggest landslide victories ever.
Only problem was, Harding's administration was far and away the most massively corrupt in all of American history and it is now primarily known for its scandals. Even the most die-hard Republican would be forced to admit that the USA would have been better off under a Democrat. Or even a third-party candidate, for that matter. But achieving that was impossible due to the previous Democrat's lies and screw-ups. (Harding himself seems to have been honest, but he was far too gullible and incompetent to do anything about his massively corrupt cabinet who were busy giving away the kitchen sink. Harding once gambled away the White House china in a card game. Once in office, Harding admitted to his close friends that the job was beyond him. That is the kind of man we are talking about.)
So plurality not only awarded the wrong winner the victory in 1912, causing immense damage, but the backlash caused by that then caused further immense damage next administration by allowing the Republicans to elect even a Harding (whereas, normally that would have at least been difficult).
1912 was like a car crash. Your head hits the steering wheel and you are dazed and covered with blood. Bad, but you figure the worst is over. Wrong: then your head slams back and breaks your neck, really hurting you.
Some think Democrat Bill Clinton might never have won the presidency in 1992 if it weren't for the candidacy of Ross Perot, who some think acted as a spoiler for George H.W. Bush in that campaign.
In recent years several Republican U.S. Senate and Governor seats were lost due to votes cast for Libertarian candidates. (E.g. Governors of: Washington State 2004, Oregon 2002, and Alaska 1982.) (The New York Sun offered their view that Bush's 2004 presidential election chances might be spoiled by the Libertarian Michael Badnarik. That fear, however, turned out to be incorrect.) In 2006, Republicans lost control of the entire Senate due to the defeat of Conrad Burns in Montana by Jon Tester by an estimated margin of 2565 votes (1%). Meanwhile Stan Jones, running as a Libertarian opposed to abortion and same-sex marriage, had 10324 votes (3%). [The libertarian also exceeded the Democrat's margin of victory in Missouri.] Similarly, in 1998 incumbent Democrat Harry Reid won the Nevada senatorship with 47.9%, defeating John Ensign(R) by 421 votes, in a race in which the Libertarian Michael Cloud got 8000 votes (1.8%).
Of course, this could cut either way. There is some speculation that really Perot was a spoiler for Clinton, not Bush. Close call. Ultimately, though, it is the randomness of plurality voting that hurts the major political parties. An excellent candidate with broad support can lose simply because a candidate with somewhat similar views also qualifies for the race and siphons off votes. That is a ridiculous state of affairs.
Also, we note that, lost in the hubbub about the Gore-Bush 2000 election and the Democrats' furor that Ralph Nader "cost Gore the election," was the fact that Pat Buchanan cost Bush New Mexico – and the critical state of Florida could easily have been added to that list. Bush won Florida by 537 votes over Gore, where Buchanan got 17484 votes. With Range Voting the vast majority of those Buchanan voters would have also contributed to Bush.
Ultimately Range Voting is about leveling the playing field for elections. This level playing field will help any political party that can produce great candidates who do good work on the behalf of voters.
Maybe you don't care about "leveling the playing field." All you care about is getting ahead. Fine. In that case, we suggest that you employ range voting in your own primaries – especially the Iowa 2008 Caucuses, since you can do that without any changes in state law, merely by changing your own internal party procedures. That way, you will get better GOP candidates. And those better candidates will get better reputations and get elected more than any Democrat candidates who were foolishly selected via non-range-voting primaries. This could make the difference in gaining the presidency, that could lead to long-term dominance – and it costs you nothing. This is one of those no-brainer moves that would cause immense benefit for your party at incredibly tiny cost, probably more benefit for less effort than any other move you could make.
Don't buy that? Check the analysis by Keech of the 1964 USA presidential nomination, showing that the Republicans, by foolishly employing plurality voting for their primaries instead of approval, Condorcet, or range voting ended up getting thumped in the general election by the biggest "landslide" defeat ever. (And here are some more studies also indicating how incredibly ill-suited plurality voting is for use in party primaries.) Do you like getting killed? If not, we suggest learning from this mistake and adopting range voting for your next presidential primary. As Santayana once said, "Those who cannot remember the past are condemned to repeat it."