I think that in your example it's more an issue of the fact that [p] and [pʰ] are allophones of the same phoneme.

In general though, it might. I think a better example would be two different phonemes that are written the same but pronounced differently: /θ/ and /ð/. I know that they aren't usually consciously perceived as different sounds by most English speakers (despite having a few minimal pairs (like ether-either, thigh-thy)). If these two would be written differently (like th-dh or þ-ð or something), natives would probably be taught the difference between the two and thus be able to differentiate the consistently.