What happens if I draw too much current from a home made battery?
Yes, you're right. There will a maximum speed that is determined by the kinetics of several parts of the system.

Electronic Resistance: Pretty straight-forward. All the parts have some inherent resistance. Though this tends to be minial compared to some of the other causes.

Migration: At high currents, the ions become depleted (or in excess) at the electrode's surface. This means that the reaction will not run at the same potential until new ions diffuse from the bulk.

Adsorption: In order for the redox reaction to occur, there is often a step needed where the ions adsorbs to the metallic surface. This is a sort of chemical reaction, so has a rate constant associated with it.

Chemical Reaction: Some batteries can also involve a chemical reaction to form an activated reactant.

Electron Transfer: This is the actual redox part itself. Transferring an electron from the metal to the reductant, or vice-versa.

All of this will play a role, but I mostly think of migration effects when I think of kinetic limitations. Once that surface layer is depleted, all of the other processes stop too.

This is most likely to be worse for a home-made battery. The electrical resistance will probably be higher because there are longer wires, less reliable construction, etc. Distances between the electrodes is longer, so in the long run there's more distance for ions to travel to equalize a charge imbalance. Your electrode surfaces are likely to be dirtier, causing slower absorption.

The book Electrochemical Methods by Alan Bard and Larry Faulkner is a good resource if you want to do a deep dive.