I wouldn't let a plug near my tire except to get me to a good tire place. 99.99999999% of places that just plug a tire don't break the tire down, because if you'd bother go through all that trouble, you'd fix it right. You didn't see a problem because the problem happened down the road after they left the shop. Not breaking down the tire is bad customer service and needlessly risks the lives of everyone in the car.
The correct way to repair a tire is:
* Remove & inspect the tire for internal damage.
This is the #1 reason plugs are bad. You have no idea if the inside of the tire is sound. This nail could have nicked another part of the inside of the tire, or there may be runflat damage that cannot be seen yet is very dangerous.
*Properly prep the injured area, then properly install a plug patch. The patch seals the inner liner and the plug seals the belt package from water intrusion. Plugs do neither, and the act of shoving the plug in can in fact further damage the belt package.
*Reinstall the tire on the rim, check for leaks & rebalance the assembly.