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... rather than calling it
"pitch-axis damping", an even better term would be "angle-of-attack
damping". An ordinary airplane in flight has
gobs of AoA stability,
marginal AoA damping, and
very little (ideally zero) sensitivity to pitch except via AoA;
i.e. AoA is a "sufficient statistic":
if you know the AoA you don't need to know the pitch.