Wait a minute. Cruz is the right wing version of Obama. That's the point. The parallels are there to see. Inexperienced back bencher barely into first term in the Senate. No legislative accomplishments. No important committee assignments. Inflexible--my way or the highway. I would throw in that Cruz, like Obama, is arrogant and smug.
The OP is "Why Shouldn't Ted Cruz be President?", not "Can Ted Cruz be Worse than Obama?"
There are no signs yet that Cruz is either arrogant or smug. He may be, but you're way out front on that one. Just trying to reinforce your reasons, I suspect.
But mostly it sounds like what you're looking for is a republican that is barely to the right of Obama. You don't have to look far. They're everywhere.