In thermodynamics, a reversible process is quasi static (because every point in the system should be in equilibrium to study the state properties and work done is maximum), but why should heat transfer occur between infinitesimally small temperature difference(why not between finite temperature difference)?
This all depends on precisely what the temperature boundary condition is at the surface of the cylinder. We can solve the transient differential heat balance equations within the cylinder as a function of time, and predict the pressure and the work done by the gas (even if the work is close to quasi static). This is a do-able calculation. But we would need to specify the problem precisely, and be sure that we are comparing apples with apples. As far as good thermodynamics books, I highly recommend "Fundamentals of Engineering Thermodynamics," by Moran et al. I particularly like their alternative treatment of the 2nd law of thermodynamics.