My guess would be that the reason potatoes aren't grown in polycultures is that harvesting disturbs the soil a LOT as you scavenge for tubers, and this would result in the other plants nearby dying. If you plant things that mature before the potatoes are harvested, crops you don't mind killing (like a cover crop maybe?), or crops that can be harvested the same way and at the same time (Sweet potatoes, oca, mashua possibly?) then it works out fine.
As far as diseases, Tom Wagner has been breeding potatoes for disease resistance for a long time, you might want to look him up. Also, many common varieties available here are, for example, late blight resistant. Ozette Fingerling is a good one.. I've heard that the common Kennebec is moderately resistant. While potatoes do have a lot of disease issues, many varieties are resistant to specific diseases, and of course it depends on what diseases you have in your area. To continue using late blight as an example; there are different strains of late blight around.
Viruses build up in vegetatively propagated plants, but growing potatoes from the true botanical seeds usually eliminates the virus problem - and possibly just keeping the plants healthy could prevent the viruses from being problematic in the next years tuber-grown plants.
Edit: Oops! Old
thread, maybe someone will benefit from these comments anyway haha..