Current in a galvanic cell
The current through a cell depends on the plate area. And on the circuit (load) resistance. And on the carrier mobility. 

And on the cell voltage -- which will depend on all of the above, which is why a cell EMF is always measured at zero current, open circuit. 

The electrode reactions certainly affect the current any cell can support, but your terms "weak" and "strong" are not clear. A "strong" electrode reaction does not mean that you have good carrier mobility through the electrolyte, and good carrier mobility does not meen that you have a good electrode reaction.

Cell current is not a chemical constant: it depends on the physical structure and history of the cell. That is the reason why it is not discussed in chemistry books.  People who are trying to make better batteries spend a lot of time working on the carrier mobility, not just the electrode reactions.