Alright. We have understood that there will be some photocurrent even when the potential maintained between the plates is zero. But is saturation current achieved at V = 0?
No, never
When the potential difference between the plates is maintained at zero, the photocurrent does not reach saturation. Never. Here's why.
Although every electron can be assumed to receive the same energy from the photons (given there is only one frequency in the beam), not all electrons use this energy in the same way. There are a few lucky ones for which the path to freedom is clear and easy, not many obstacles, not many collisions. When these electrons make it out of the surface, they retain a good fraction of the energy received from the photons.
But there are others for which life ain't easy. Freedom from the metal surface is not even guaranteed. These electrons undergo multiple inelastic collisions and lose a lot of energy; if they have any energy left by the time they reach the surface boundary, that gets converted into a small amount of kinetic energy, which can sometimes get very close to zero.
The Struggle is real
1. Some of these low energy electrons do not make it to the collector plate, and generally get accumulated near the emitter plate. This is one reason why the current recorded at V = 0 is lesser than the saturation current
2. This accumulated concentration of negative charges (the slow electrons - their combined negative electric field is represented in blue) right outside the emitter plate, sometimes called space charge, also creates a resistance for the newly ejected electrons due to electrical repulsion, thus further
slowing some fresh electrons down
These two effects combined tells us that in the absence of an accelerating potential (i.e., at V = 0) the photocurrent measured is less than the saturation current.