It's not uncommon with a failing HD for it to crash after a complete reinstall of the OS. It just seems to push it over the brink. Logical when you think about it. If it's the original drive then it is 8 years old which is well past the average of 5-6 years years for a HDD.
Personally I'd go with the replacement SSD, my wife did this on her 2011 MBP some years back and the improvement was very noticeable. I think using an external drive will negate the speed advantages of an SSD but it would get you up and running.