At 50A (guesstimated current for all 2400w of controller draw), it would mean the 13Ah battery has to be capable of at least 50 / 13 = 3.8 or about 4C. Most of the cheap ones can barely do 2C (or twice the Ah as current Amps), which for this one would barely be half of the required current, at only 26A, and that's usually pushing them hard. Without knowing what's actualy in the battery, you can only hope it can handle the load, and if it can't replace it with one that can.
If it was actually used in conditions that continuously drew 50A, it would be empty in less than 15 minutes. It would probably shut down before that from voltage sag under load, probably around the 10 minute mark. It's unlikely that it could be kept at full power very long unless they ride it up long hills or against strong headwinds, or ride fast enough.
if the controllers smelled like smoke, they certainly could have severe damage--the most common is FETs blown. But if those are blown, a common reason for that beyond simple overload is a motor problem or wiring between motor and controller, either of which will then blow up any new controller that is hooked up.
I'd recommend at least a minimal investigation into the inside of the controllers, to see what actually blew up and smelled smoky, and/or verifying the motors and wiring don't have any problems.