For an oceanic/ETOPs flight it is not safe and I would have thought not legal.
Not true. Depends on the aircraft.
A B777 can operate ETOPS with an inop APU, but does have two generators on each engine.
The problems with inop APU are mainly cabin cooling. When I worked for Gulf Air we had a house rule. With APU inop you tried to fix it on the first nightstop. If you failed, you changed it on the second.
The problem here in the frozen North is the difficulty in pushing back an aircraft with engines running over an icy ramp.