2. Homeschooling academics can be more rigorous. As an engineer, I consider math to be the foundation of all my success, and common core has turned math into a laughingstock. Enter homeschooling, where I can pick the "Singapore Math" curriculum. Singapore typically scores number 2 every year on the international math achievement exams, their math program is entirely in (British) English, and I can have their exact program for my kids instead of common core.
Losing mod points to respond to this, but as a math teacher I can say based on substantial study and practice that you are drinking the wrong cool aid about common core. To see why you have to understand what common core is and isn't. Common core is just the set of standards, and it's important to read a bit of them to realize what that means and doesn't. Everything else that isn't written in the common core standards, but yet people still incorrectly call "common core" is just an attempt at implementation of the common core. If you see a given incomprehensible homework assignment, that's the implementation, not the common core. The standards don't give implementation details, the teacher, textbook, and/or district provide those.
Before I go further though, I will agree with your first statement. Homeschooling can be more rigorous, but as I am considering homeschooling my children, the thing I would do would be to implement the common core to it's fullest extent and attempt to exceed it.
I'll cut to the chase and give you the summary: the common core standards are more rigorous and are a substantial improvement on every state standard before them that I am aware of. They embody the important parts of the best of education research and the math standards for example are substantially based on the previous work of the Principles for School Mathematics developed by NCTM. So if a given implementation is bad, it means either the teacher or textbook are not as good as they could be, not that common core is bad.
Look at the Standards for Mathematical practice:
and the following specific standard:
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A classroom attempting to implement the standards for mathematical practice to fully meet the standard above will be leaps and bounds above a traditional classroom in terms of rigor, cognitive demand, ability to reach diverse students, etc.
Also having studied the TIMMS study in depth I can say that the reason Singapore does well on the exams isn't necessarily related to their curriculum, but likely has more to do with parental support for education.