So, you can pump water out of the steam wand but not through the group head? That would either be a grouphead blockage or a 3-way, but you've already checked the 3-way.
You've already tried removing the grouphead. I'd maybe try letting it run in steam mode, and then switch the brew button on. This blasts steam pressure through the grouphead (assuming it isn't still blocked!).
So, you can pump water out of the steam wand but not through the group head? That would either be a grouphead blockage or a 3-way, but you've already checked the 3-way.
I'd use bottled water. Easier, and better coffee.