Infinitesimal and Heat Sink/Bath
Infinitesimal arguments are well suited to many forms of calculus. However they can lend themselves to mental hardship when simply applied to thermodynamics and are not given context.
Certainly infinitesimal changes tend to be reversible, but such infinitesimal changes are NOT measureable because they are so small. Moreover they are often meaningless because true reversibility often only occurs when the infinitesimal change is zero, or if prefer reversibility more closely approaches reality as infinitesimal change approaches zero.
We also realize that if you had a sufficient/massive number (approaching infinite) number of infinitesimal changes then the change can become real. And if this change involved work or thermal energy change then such real change is generally NOT reversible.
Similar arguments apply to heat sink or heat baths. In these cases real change often does not affect the heat sink/baths measureable parameters. In other words real energy change tends to be NOT measureable. This is not to say that the change is infinitesimal but the principles are very much the same.
Again, if you had a sufficient/massive number (approaching infinite) number of real changes then the change in the heat sink/bath will become measureable.
And the mother of all heat baths/sinks that we deal with on a daily basis is the Earth’s atmosphere.
Never forget whether thinking infinitesimal change to real systems or real change to a heat sink/bath, that the change cannot be simply ignored unless it is zero, in which case it is completely meaningless.