Sure, 40% of voters claim to be independents, while 31% claim to be Democrats and 27% claim to be Republicans. Based on those dumb stats, it would appear Republicans need independents.
But here is what those stats do not show - 40% of all voters claim to be CONSERVATIVE (as opposed to "republican"). Meanwhile, only 18% claim to be liberals. So, conservatives outnumber liberals by a wide margin.
So here is the untold truth - Republicans do not need independents. They need CONSERVATIVES. If the Republican party decides to actually stand for conservative values, they will have the support of the Republicans AND the conservative portion of the independents. And that would give them a majority, since most "independents" lean to the right (remember - conservatives outnumber liberals by more than 2 to 1).
In 2008 Republicans ran a moderate RiNO Republican in the hope of attracting independents, and Republicans lost - McCain was not conservative enough, and conservative independents just stayed home on election day.
No, Republicans do not need to compromise their values in order to court independents. On the contrary - they need to be conservative enough to get the conservative majority fired up!