Moment can be calculated with respect to any point you choose, since you may fix any point you want to check how things rotate about it. You just need to be careful with your interpretation of what that calculation means.
Indeed, with respect to the point of contact with the ground, the tyres don't create a moment, but they are not the whole story.
As an analogy, think of a bar connected to an axis such that it can be freely rotated. When you push the bar to rotate it, you create a torque with respect to the axis of rotation. But when you switch your reference point to the point of contact of your hands with the bar, you will feel the force and torque that the now rotating axis (it rotates around you in your rest frame) applies on your hands, and these will be the relevant quantities to use in your diagrams.
Going back to the bike, when considering a rotating reference frame, you need to take into account the centrifugal forces that emerge in that frame, which in this case arise due to fact that the upper part of the biker+bike system "tries" to move in a straight line with nothing but gravity to balance it.