Back when crushable foam helmets first came out, there was just an elastic cover wrapped around a foam helmet. The effect of the helmet impact was that it always broke apart; the helmet cover just kept the pieces together after impact.
Most helmets now have a plastic skin over the foam which provides some structural integrity while the foam is being crushed [1]. But you are correct, the helmet only works because the kinetic energy is being absorbed by a material whose composition (closed cells) will naturally break apart into smaller pieces when enough kinetic energy is applied.