I forget if the Vibrant is a true TBE, or if it bolts to the s-pipe?
I've installed two true TBEs, and both times I drained the coolant and yanked the radiator--in which case overheating could absolutely be related to the exhaust install. If it bolts to the s-pipe, then disregard the above... lol
Explain the symptoms a little more... you say it overheated? How high was the temp gauge? Did you open the radiator cap (after the car cooled down) to check coolant level, or did you just look at the overflow tank? What kind of temps outside? How hard were you driving the car on that 10 min drive?
In regards to the failure to start, does it crank but not fire, or does it not even crank? If it doesn't even turn over, do you hear a click or anything when you turn the key? All of your electrical accessories work fine? How did it die--did it lurch to a stop suddenly, or did it sputter and die?
I'm not seeing the failure to start being related to the overheating in a mechanical sense--at least not in the sense that you seized the motor or anything. One thing that would make sense is if you broke the inner accessory belt that drives the alternator and the water pump. If that belt goes, you lose your ability to charge the battery or generate power, and the ability to circulate coolant--that's both your symptoms right there. Check your belts--there should be two of them. And try jump-starting the car as well.