Not only is md5sum not proven to have equal dostribution, it is specifically known to not have equal distribution, only nearly equal distribution.
A hash function is any function that converts an arbitrary input size into a specific output size deterministically. No other requirements are there. A hash for a simple job could be just adding the ascii values together and give the output. Needless to say, that would not have an even distribution.
Linux? No. Proton, yes, unless they are using kernel-level anticheat like Apex