Best Path for PhD in Engineering - which Undergrad Major at which Colleges?

No it is not true. Major in electrical or mechanical engineering depending on which you prefer. Robotics is some combination of the two. You will have plenty of math and physics. Secondly, I you want a PhD, I would worry less about avoiding debt and more about going to a good competitive school where you can engage in research. If you’re from Michigan, this is an absolute no brainer. Michigan has a great program for undergraduate research.