Quasi-static implies that T aprox equals Tsurr?
In my judgment, when they are talking about irreversible cycles, they are referring to idealized cases in which (a) the process is internally reversible with respect to the working fluid but (b) irreversible with regard to the region outside the working fluid where there is a finite thermal resistance (e.g., a cylinder wall with finite thermal conductivity) between an (external) ideal constant-temperature reservoir and the working fluid.  So the rate of heat transfer to the working fluid is very low (quasi static), and all the entropy generation takes place within the (higher) thermal resistance wall.  At least, this is how the irreversibility is often modeled.

To illustrate all this, see the following thread from Physics Stack Exchange:  https://physics.stackexchange.com/questions/443527/entropy-generation-in-an-endoreversible-engine