Yes, you are right! In fact, to people working in foundations of mathematics it is routine to compile away any explicit use of the golden ratio in Conway's proof and thus obtain a proof which only refers to the integers. However the resulting proof will be less perspicuous. I don't know of any transparent proof avoiding the golden ratio, though it definitely could exist.
Strictly speaking, it's Conway's proof uses the golden ratio. But it could be that there's an alternate proof that doesn't use the golden ratio.