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Humans have problems with this - sometimes pilots get disoriented, and think they are flying straight and level, while their instruments tell them that they are in a spin. Statistically, the instruments are more reliable than the human sense of balance (especially at night or in fog/cloud or turbulence).
As far as the A320, the autopilot should be able to calculate things like lift, velocity, and wing turbulence as part of optimum climb calculations in all situations by evaluating real-time data.
How long were the pilots tracking the storm that they were headed into? A 2-minute window for a critical maneuver sounds quite short for a storm that should have been visible from the radar before the plane even took off.
It ought to be possible to use GPS to determine how the plane is moving instead of relying on sensors which can be iced up, although there will still be difficulties when there's a high wind speed (which may be a powerful downdraught in a storm) or if you're flying through the jet stream.