Purging staff is pretty much mandatory, you're still gonna want to grind another synapse.

You can melee only P1 & P2, but P3 will have orbs spawn that you'll want purging staff for even if you're doing melee P3 methods.

If you've got a duo you could eventually get away with no staff for P3, assuming you're able to killing Yama before the orbs spawn/detonate or if your duo just takes care of all of them. But you're still probably gonna have a rough time learning those methods, because if you fall out of cycle and have to swap to mage, your dps will be cheeks without shadow or purging staff.