I understand how Carnot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it is less clear why they need to be less efficient, as I have seen stated in some places.

If they could be equally efficient then an irreversible engine engine could be used to drive a reversible engine operating between the same heat reservoirs, without any net energy transfer between the reservoirs. It would then be unclear what is irreversible about the irreversible engine. Does that constitute an actual contradiction though? If so can the argument be stated more tightly? It feels a little sloppy as is.

It could also be a question of how an irreversible engine is defined in Carnot's theorem. I understood it to mean one that cannot be run in reverse as a heat pump, which could presumably include a Carnot engine with a one way ratchet attached. If it really means an engine whose thermodynamic effects can not be undone, the implication would be trivial.