Does the reset button need to be hooked up?

According to the Jet 3 manual
Pride Mobility Products Corp Product Owners Manuals Specification Sheets (It's in the maintenance chapter) there is a decal in the battery compartment with the wiring diagram on it.
But yes, that breaker needs to be in the circuit.

The reset button alerts you of an over current situation. When you are pulling too much current, the reset button will pop. This could indicate a short circuit in the wiring. Without this in the circuit the wires could get very hot and melt the wiring. Worst case, it could cause a fire. If you are confined to a chair and cant get out, this could be bad. This was placed in the circuit to prevent excessive damage to the chair. If you need to replace it, please contact a service tech to look at the chair. You should never have to replace this component.

Yes of course if it was made with the unit its was important then as
well as it should be now contact manufacture. Reset buttons reset micro processor functions as well as reset the unit memory capability.. Hope this helps God bless