"the guy should be booted out" - On the simple basis that you'd like that? In any event, if you choose not to block someone's PMs, then that's on you. Don't go crying to Admin if you continue getting them.
"I think board mods should have term limits" - They don't.
"what actual purpose do they serve to admin?" - Board Moderators generously volunteer the time they have to keeping the boards they moderate iBox updated, add sticky notes and remove TOS violations when they occur.
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