The importance of reversible processes(?)

In thermodynamics(at least in classical thermodynamics),the idealization of reversible processes is used time and again.Can we say it is central to thermodynamics?
I mean can we say it is so important that if we can't find a reversible counterpart for an irreversible process,then the existence of that process is in contradiction with thermodynamics?If yes,can we just drop the notion of reversible process in favor of the existence of one having no reversible counterpart?
Thanks