put my old hard drive into one of these enclosures, obviously purchasing one of these from the shop.
Basically yes.
But the problem you have is really simple, I am slightly concerned about anyone acting as a repair shop yet can't fix such an issue with ease, it is a simple case of swapping out bits until you find the culprit, which is only difficult if you are not sitting on a pile of spare parts (and I understand like yourself are not used to pulling them apart).
Your internal hard drive just sits in a bay with 1 power cord and one data cable attached to it.
You can unplug it, and insert it into a hard drive enclosure, which allows you to plug it in via USB to any other computer, and then just read the data like you would a memory stick.