I'm not at all convinced that the issue is whether you can hop between an airliner and a single engined piston aircraft. As previous posters have indicated, you can also hand-fly a transport aircraft. I think that the problem is the gulf between highly integrated information rich modern cockpits, and remembering that it is "just" an airplane, and behaves exactly like every other aircraft in terms of pitch/power/lift/drag and all those simple aerodynamic laws.
Many studies have looked at the surprisingly poor job pilots make of the transition from VFR to IFR flight. Even experienced instrument pilots tend to "hang on" to fast disappearing visual cues rather than simply transferring their gaze to the instruments and flying the way they were trained. The number of continued VFR-into-IFR accidents where both the pilot and the aircraft were fully IFR capable is testament to just how hard that can be.
In my experience, the same applies the other way around. Many times, with new-ish first officers (and some not so new), I was surprised at their inability to transition from IFR flight to a simple visual approach. On a lovely day, their sector, we would brief for the instrument approach, and if it was a smaller airfield where a "shortcut to a visual" was possible, we would brief for that as "plan B". Sure enough, the airport is right there, the pattern is empty, and ATC offer the visual. We accept, but for some FO's this became a stressful moment! They were so strongly oriented to commanding the FMS or A/P through bugs and buttons that they found "flying the aircraft" to be quite overwhelming - EVEN THOUGH THEY HAD THOUSANDS OF HOURS HAND FLYING OTHER TYPES.
In other words, the problem is not about basic muscle memory, or experience, it is very situational. Humans are poor at switching context.
After 4000 hours in the Airbus cockpit, the AF pilot would be completely baffled by the contradictions being presented, and in the 3 minutes remaining probably never figured out what was going on. Paradoxically, a pilot from almost any other type of aircraft might have spotted in a few seconds that the aircraft was fully stalled, and be able to solve it.