.. and the other half (ossur lehistakel) your conveniently forgot.
And just because you have not seen it doesn’t mean it’s not a Rebbishe minhag. Are you Rebbish? Have you been to all Rebbish chasunas? Actually I have not either, (I’m not Rebbish, they for some reason don’t personally extend their invitations to me), I have seen pictures / clips of countless Rebbish chasunas where the Kallah covers in the same manner.