I have a glass top induction burner and a glass top electric range. I shake my pans all the time and have never scratched either and have been using both for half a decade. I can't imagine putting anything underneath my cookware to try to spare the cooktop or allegedly make cleaning easier.
Update: Okay, so I looked at my cooktop closely and there are some fine scratches on the most used burners. They’re not immediately visible unless you’re really looking for them, and I’d consider them to be part of normal wear and tear, especially given how much the cooktop has been used (and how much shakin’ has been goin’ on).
