In Newton's pendulum, the kinetic energy of the first ball is transmitted through all the balls to the last because they are perfectly elastic and very hard. If the balls were perfectly inelastic, the last ball would not move at all because all of the balls would have absorbed all the energy.
In a hard tire, all the energy is transmitted through and you can feel all the vertical bumps. Not only do you go up and down, you go forward.
In a soft tire, the rubber absorbs all the energy and reduces your forward motion. Try riding on grass and sand where the soft ground is absorbing all the energy. Then compare that with riding on concrete, asphalt or on a perfectly smooth and hard track of a velodrome. The hard surface transmits all your energy.