Actually, Republicans LOVE the idea of eliminating 'Winner take all' electoral rules for each state...but only in states run at the state level by the GOP but which generally vote Democrat at the national level (like Michigan). In states that reliably vote red overall, they oppose eliminating the 'Winner take all' rules tooth, claw, and nail.
Gee, I wonder why?
If you really want to get to 'every vote counts', don't go to districts - all that does is disenfranchise democratic voters, since liberals tend to live in urban areas, while rural residents tend to be conservative, and right now more states than ever are gerrymandered to an extreme we haven't seen since the 1800's.
Just remove the electoral college entirely and have the presidency determined the way states choose their senators - pure popular vote. Right now, numbers do favor Democrats SLIGHTLY, but that goes back and forth over the years, and there are more than enough moderates that they're the ones who end up actually deciding things. (And for the record, "a few cities would run the nation" is pure sophistry - the largest ten cities in the US combined account for less than 10% of the US population (24.8 million out of 330 million), and not all of those ten lean liberal, anyway. The purpose behind that argument is to scare people into thinking Democrats would take over forever, and it tricks people into thinking that acreage should be factored into voting power.)
Even with that, however, we're stuck with the two-party system as long as the Presidency is determined by majority vote (as opposed to plurality). With the majority vote system, it's basically demanded that there only be two sides to the election - any more than that dilutes one side or the other, guaranteeing that they will never win. In a plurality system, however, there's much less encouragement for people to form two monolithic parties, since all that is needed is the highest number of votes. That would actually allow more people to support other parties, and tend toward reducing the us-vs-them mentality we have here now. (Hell, the biggest reason I vote Democrat rather than Green or for some Independent is because that at anything beyond the local level (with a few exceptions like Sanders), there is literally no chance for someone whose name isn't followed on the voting form by either R or D to EVER win the election.
But as Mark_ said, that won't happen in the US. Neither the GOP nor the Democrats will ever allow it to happen, no matter how well it works in the rest of the world. People virtually never voluntarily give up political power - that's one thing that made George Washington and Cincinnatus so special.