This entire thread is basically tl;dr, but I'd like to propose an alternate solution of my own.

Make accessories not contribute towards item level at all.

In the vast majority of cases where undergeared players cheese item level requirements to get into dungeons that they have little to no chance of clearing (or expect a carry), it's done through equipping accessories that don't even belong to their class, because accessories have no class restrictions. And accessories also happen to be the easiest high item level gear to get. In such cases, it's BLATANTLY clear that the ONLY reason they're equipping such accessories is because they want people to carry them through something they shouldn't have any business doing in the first place.

This type of solution will basically ensure that players WILL actually meet the statistical requirement to make a meaningful contribution towards dungeon runs, if only because the weapons and armors have much tighter class restrictions to equip them.