"Force schools to compete" I don't think this is it.
'Competition' is definitely not the right word for education.
The best education systems in the world: S. Korea, Finland - are not competitive in the least.
If you have good students (re: good parents), decent teachers, a classroom, textbooks and a chalkboard - you're going to get a good education.
The 'choice' issue I think is important when it comes to alternative and special interest schools - in particular those in hard-up places: they have Charter Schools with very special types of programs geared towards kids from rougher districts etc.. Or maybe an ethnic/religious slant i.e. 'Ukranian school' or 'Catholic School' - neither of which would be entirely different from a public school, other than at a 'Catholic' or 'Ukranian' school there would be a 'Christmas Concert' instead of a 'Winter Concert'. But still, culture is important.
Forcing schools into heavy competition I think will just have them focus on the wrong things, marketing themselves, inflating numbers, managing perception - and that's wasteful.
And as long as the 'common core' is not too onerous - I think it's probably ok as well. I mean, who'd be against basic reading, writing and arithmetic?
Get rid of Common Core. Force schools to compete. Now, every schools will have to teach their students better.