If the problem is so widespread I imagine that all the really rich and famous have A few people on staff to filter incoming correspondence, triage and follow up on credible threats, check with police as needed etc.

What of the modestly famous and not that rich? There ought to be an agency where one could outsource this sort of work for a fraction of the in-house cost. The agency would accumulate experience and might even keep the database of known crazies which gives it economy of scale.