On the 727 if the APU is out, you're dead in the water, so to speak. You cannot start the engines or refuel the aircraft unless ground power, GPU, is available and you have an ground air source to start at least one engine.
In fact, unless modified, if you leave an engine running after landing, you still cannot refuel with out a GPU on the 727.
Oh, the APU on the 727 cannot be used in flight, ground operations only.
That's the furthest thing from the truth I've heard in a while. Sentence by sentence: You can start the engines off Battery Power only, the aircraft can be refueled by manually opening the fueling valves in the wing (#1 and #3) and under the LE devices on the right wing (#2). Ground Air/Start Cart is needed though (GPU not required).
Unless Boeing went the several versions of the fueling valves and manifolds (which they didn't) the 727 can be refueled without even the battery being on. The only part you got right is that the 727 APU shouldn't be used in flight (it can't be started anyway, but can be left on accidentally after takeoff, and will flame out around 25000 feet), some operators had the T/O Warning horn attached to the APU in addition to the Flaps, Trim and Speedbrakes to prevent inadvertant takeoff with the APU on.
Bar none it's one of the best remote operations jets in the world.