I think the issue with this is that a lot of people in the USA who thought of themselves as firmly "on the right" prior to the events of 2016/2017, will now be forced to choose "neither" because they are disgusted with the current administration's impending meltdown, even though they still consider themselves the opposite of the "left".
Then you have people like me; I'm not a centrist (I don't think), but I have views peppered across the spectrum. I am pro-choice in reproduction rights, I am firmly and proudly pro-2nd Amendment, I believe strongly in LGBT equality, religious freedom for all religions (even the ones that give me the shivers), I feel there should be an easier but at the same time more secure path to citizenship for undocumented immigrants and legal immigrants alike.
I believe in small, efficient, non-intrusive government that serves all the people, not just select groups and ideologies. I am vehemently against government surveillance of its own citizens for any reason. I think getting rid of Obamacare is stupid; improve it, don't scrap it. I think we should reduce our military presence in countries that don't want us policing them, yet stand ready to defend countries that truly need our help when they are attacked by their enemies. I believe the 1st Amendment is vital to the continued freedom of our nation, and I despise our current president for attempting to chill free speech and free press.
I believe if you're going to give tax breaks to small businesses (which I think is a good thing to do), you need to also raise the minimum wage so some of those savings can go towards improving the quality of life for employees of those businesses.
So what am I? I'm "right" on a few issues, "left" on others, and somewhere else entirely on yet other issues. I voted Libertarian again this term because it's the closest to my own values, yet I don't agree with everything Libertarian either.
In short, I think you need a few more options than just Left, Right, and Neither.