OVERTALK:
If the APU hadn't been started, the degree of controllability that led to a "sufficient" flare (although not optimal - NTSB Report) and subsequent benign ditching might well have been characterised as a crash.... with a significant loss of life due to a greater loss of airframe integrity.
The APU supplies only pneumatics and electrical power generation on ground and in the air. (On heavy takeoffs, one can run the packs off the APU to reduce engine bleed and slightly increase power while keeping temperatures slightly lower.) Starting the APU is a standard procedure when an engine is lost so the reaction was a good one given the circumstances.
Here, the engines were still rotating slightly - 35% and 17% if I recall from earlier in the thread but I may be wrong on the numbers - the point is, both hydraulic and electrical power generation can occur at these N2 speeds.
The RAT would not deploy under these conditions as at least one if not both of the two AC buses was powered.
The RAT only powers the "Blue" hydraulic system which would power only the slats. The minimum RAT operating speed is 140kts, so there is another consideration in terms of control and available hydraulic power; they got Flaps to Config 2 so there had to be "Green" and/or "Yellow", (vice 1, 2 or3, or left/center/right), hydraulic system power.
With the RAT extended, an emergency generator driven by a hydraulic motor powered by the blue system can supply limited electrical power. The airplane would be in Alternate Law and there would be limited instrumentation. The possibility of it dropping to Direct Law exists as speed is reduced and hydraulic power lost. At this point it gets a bit messy.
Under the circumstances experienced by this flight, with electrical power the flight control system would have remained in Normal Law and the APU would be providing backup "just in case", rather than powering the aircraft electrical system but in the time available and priorities considered, the crew would almost certainly not make a decision to rely upon windmilling engines.
If the engine(s) had not been supplying electrical power and the emergency generator was unable to supply electrical power, the batteries would supply sufficient power to support the flight control system. The RAT would power sufficient hydraulics to move the necessary surfaces but the flaps could not be extended, only the slats, as described above. It might have turned out slightly differently - bit faster; I believe touchdown was around 107kts, but again that's just from recollection - someone else will surely know and help out.
Further info: On battery-only flight, the battery is certified for 30 minutes but I would not trust that length of time. On an A320 series aircraft, if the APU or the APU generator is u/s and one loses an engine or engine generator in my opinion as a captain that is an emergency and a landing at the nearest suitable airport. The ECAM - LAND ASAP would be an amber, not red message.
The A320 is emminently flyable in Alternate or Direct Law. It becomes a "WYSIWYG" airplane without any of the autoflight system protections. It is not a problem to land in either mode.