Why do you doubt the explanations? I think the vibration talked about here is before automatic braking and gear retraction. I think it is worst when the wheels are unloaded off the ground and rotating very fast in unloaded bearings. Whilst the noise can be alarming when seated near to the undercarriage legs, with the general noise of the engines, just a few rows away you cannot hear it above the rest of the noise, and that explains the unconcern from the crew away from the wheels. If you run pushchair wheels very fast, then lift them off the ground, they can suddenly vibrate quite badly when unloaded from the ground. that is what is happening to the aeroplane wheels.
As for gyroscopic forces on the APU/engines, not a factor. They are designed to take it. Indeed, these forces were looked on as one possible cause of the Comet accidents in the 1950s and shown to be irrelevant. Aeroplanes are designed to manoeuvre with the engines at high power and the APU running. The actual swing has no effect on gyroscopic forces anyway, it is the angular change and rate that counts, not sideways or vertical movement.
With flap out, you can see marked vibration and shaking on the flaps, but at take-off, lower settings are used than on landing, and the shaking is in excess of the shaking on mid-flap settings. Again, not a factor I believe.