They should do both, set minimum ilevel AND level sync to whatever content comes after it. (eg The minimum ilevel of HW should be the ilevel 120 Poetics gear, the minimum ilevel of the SB dungeons should be the poetics ilevel 260 HW gear)

To do otherwise makes the leveling content absurdly easy when overgeared, it seems ridiculous that all 50/60 dungeons weren't ilevel synced when SB came out.

EG:
Sastasha to The Aurum Vale should have ilevel sync at 55, and minimum ilevel set the same as the dungeon level - 10.
The Dusk Vigil to the Great Gubal Library should ilevel sync 130-150, with a minimum ilevel at 120 (same as ARR Poetics gear/World of Darkness gear)
The Sirensong Sea to Castrum Abania should ilevel sync to 270-290, with a minimum ilevel at 260 (Same as HW Poetics gear/Dun Scaith)


But this doesn't solve the actual problem of requiring capable gear in all left-hand slots, or relevant materia melded to that gear. It needs to be said again, if you can enter the dungeon, you can do the dungeon, you are not promised a faceroll every time, and if minimum ilevel is enforced, so should level sync.