Why does more heat transfer take place in a reversible process than in a irreversible process?
In the Clausius inequality, the temperature in the denominator is supposed to be the temperature at the boundary of the system where the heat transfer is taking place.  This important point is often omitted from many thermodynamics textbooks.  In a reversible process, the temperature at the boundary is essentially equal to the temperature of the system, because the system temperature is uniform throughout.  But, in an irreversible process, the temperature of the system is not uniform, and the temperature at the boundary can differ substantially from even the average temperature of the system.  So the Clausius inequality does not always require that $\delta Q_{rev}\geq \delta Q$.