The technology is there already to have no pilots..
. . . Yes, as long as it's a normal flight and everything works. No hyd/fuel valve/pump failures; no A/P, A/T snafus; no auto press failures; no bird strikes; no asymetrical/split flaps; no smoke in the cabin . . .
Check with the military and ask how many multi-million-dollar UAVs haven't returned to home base.
As to Captain Birkill: Where is BALPA? Where are his BA colleagues? BA should have him fly right seat for 6 months and let him get his bearings back, then have him resume his captain duties.