Consider what happens when you've just turned the thing on and it has just barely expelled its first proton through the negative grid. As long as the proton is

Consider what happens when you've just turned the thing on and it has just barely expelled its first proton through the negative grid. As long as the proton is close to the grid, it may as well be on the grid: it cancels out the contribution of one electron to the field between the plates, which means the voltage across the gap is incrementally reduced. This reduction in voltage allows one electron to flow out of the generator into the grid. Of course, as the proton moves away, its contribution to the field between the plates diminishes at a rate depending on the ratio of its velocity to the radius of the grids. (This is why the infinite-grid case is easier to think about: the contribution doesn't diminish at all.) Eventually — and depending on the proton velocity and grid size, "eventually" might mean milliseconds — its contribution becomes negligible. At that point a steady state will have been reached, and (as you intuit) it won't be possible for the generator to push any more charge onto the grids without increasing its output voltage.