That isn’t how US politics works. Republicans don’t move their position to match voters preferences they endeavor to keep their base separate and distinct, so they firmly identify as Republicans, then use that identity to tell their base what positions they should hold.
Another way to think of it, which is mostly just a different way to express the same thing:
There are swing voters, but the ones in the middle don't count for as much as commonly expressed. Many swing voters out there don't swing between Republican/Democrat. Instead, they are further from center, and swing between Republican/Libertarian/Constitutionalist or between Democrat/Green.
Closer to the middle, there are some people who go back and forth between Republican and Democrat, but they are outnumbered by those who just go back and forth between vote/don't vote, but can re relied upon to be pretty reliable with which way they do vote, assuming they can make the effort to do anything at all.
In this situation, enthusiasm is key. The Republican party has had success for many years by moving rightward, getting the less moderate people enthused. The Dems just can't seem to match the level of enthusiasm that the Republicans have, perhaps the far left is just too distrustful of authority to really buy in to the Democratic Party as a whole or maybe the Dems just need to do better messaging. Add in the effect of gerrymandering (which currently mostly favors Republicans) and the over-representation of rural areas in the U.S. and most state Senates, and you've got a strategy for the Republican party to do very well by strongly motivating a relatively small set of voters
So moving left/right just does not have much value, too many voters are siloed together. Obama said something that Republicans think does not sound very Democratic Party-like. Obama is still pro-choice on abortion, he still favors more regulation of firearms, still supports a much stronger push towards renewable energy and much greater regulation of emissions. He still supports efforts to make it easier to vote, still supports a smaller military budget than preferred by the GOP, still favors full gender equality. Death penalty, homosexual people in the military, the list of issues with pretty clearly defined differences between the two parties goes on. Get the idea? - these, in many cases are almost black and white issues relative to the current status quo. We increase regulations or restrictions on this or that, or we oppose new regulations. We want government incentives to make this or that change, or we don't, more funding, or less.
And most voters fall into these silos. Find a person who strong favors continuation of the death penalty, that person will likely also oppose any new gun regulations, oppose abortion, and not be supportive of increased regulations on emissions.
This used to not be the case, there used to be issues that didn't fall along party lines. But there are fewer and fewer such issues, making it harder and harder to parties to get people to switch votes.
