I understand an observational study is the best you can get, but self-selection on mushroom consumption seems an impossible to correct variable. My stereotypes would expect elevated-mushroom eaters to consume more raw food, salads, etc.
There are also studies that show the possible mechanisms behind the effect. One is PSK (used in Japan and originally isolated from shiitake) and another is turkey tail. Both are used to treat cancer by improving immune response in conjunction with other treatments.