I won't say you're wrong since I stopped taking physics in high school, but this makes absolutely no sense to me. How does defining the Planck speed unit to be the speed of light suddenly make c a unitless constant? It's not "c=1" (except in informal shorthand), it's "c = 1 Planck speed unit". It seems like all sorts of things would "break" very quickly when you start dropping units.

Maybe someone has a link that goes into more depth on why all of this is okay?