The issue that I have is that if you could put the wheels on weighing scales and measure the weight upright and then compare it with the weight when the bike is at an angle, then they would have to be the same.
Not so, I think. "Weight" is actually a synonym for "force" in this case, and is defined as mass times the force of acceleration (in this case, gravity). It is a mistake to conflate "mass" and "weight". That is why you weigh less on the moon: your mass is the same but the moon does not have as much gravity so you exert less force on it.
Mass is a constant but as I think we have just agreed the "amount" of gravity that gives the wheel (and the entire assembly of rider and machine) its weight is diminished; some of that force (weight) is no longer directed at the contact patch.
In the extreme case where the bike is lying on its side, as for example after a tip-over, would you agree that the tire is in fact exerting zero force on the ground at what was the contact patch when the wheel was upright? The mass has not changed but none of the force of gravity is being exerted to hold the tire (at the contact patch) to the ground.
If you could lean the bike and measure the force exerted by the tires (or "tyres" as you traditionalists spell it..
) while riding in a straight line you would, I think, see a diminution of the force of the tires on the ground.
Adding a change in direction brings into play another set of forces, parallel to the ground rather than perpendicular to it. When that set of forces exceeds the downward force of the effective weight plus the intrinsic coefficient of friction or adhesion between rubber and road, bad things tend to happen...