half-reaction in a battery
The half-reaction stops primarily because it's still producing/consuming electrons, and there's only so many free electrons an electrode (the local environment) can accept/supply.

Just like chemical species, electrons have an equivalent to concentration. (Whereas the generalization of the concentration is the "chemical potential" of the species, the "concentration" of electrons is related to the electrical potential or voltage). If the "concentration" of electrons gets too high or drops too low, the reaction rate of the half reaction will drop off - just like it does if the concentration of a chemical product/reactant gets too high/drops too low. In normal operation, the external conductor allows electrons to move, renormalizing the "concentration" of electrons and allowing the reaction to proceed further.

Note that if you have a poorly made half-cell, you may indeed have a pathway for the electrons to travel to the other half cell, or to react in some other (half-)reaction. If this occurs, then your battery will indeed continue to run even when disconnected, draining its stored chemical energy. But decently made batteries control for this, and the only way for the half cells to receive/dispose of electrons is through the external circuit.