when I add all possible frequencies I get a wave that has no wave because it cancels it out
Photons can behave like waves, but they don't combine the way you are trying to combine them.
These little bundles don't combine mathematically. They don't merge into a single wave. They are independent little bundles of energy.
Isn't one of the most readily-observed consequences of wave-particle duality that "waves" of light can, in fact, exhibit interference?