I've always thought this was a good idea. The idea of the penalty shot is to give the opportunity back to the shooter that the penalty denied. I think you can argue it provides a better opportunity since he gets to skate in unchallenged by pursuing defenders. But be that as it may, the penalty still did occur, so if you restore the opportunity and it leads to no goal, enforce the penalty, just as you would on a random hook or hold. In essence, the penalty shot just allows the play in which the foul occurs to conclude.
I would not enforce the penalty when a goal is scored on the play or the penalty shot. My rationale is the "harmed" team got an advantage from the penalty by having a 6 on 5 situation (or a penalty shot), and that resulted in a goal. No point in penalizing the offending team twice when a goal already resulted.
The big problem with this is that it then becomes significantly more advantageous to dive and embellish on a break away than to actually try to score. You end up with the original hampered opportunity; if you don't score on it you get the PS; if you miss on the PS you get a powerplay. That's overkill.
The penalty shot is perfect. It's the only disciplinary intervention in the sport that is completely fair. Got unfairly hindered during a prime scoring chance? You get another, even better one right back. Generally I see way less complaints and frustration about penalty shots than I see about every other penalty call. No reason to make it worse by adding punishment for punishment's sake.
I'm always baffled that people continuously complain that referees are terrible, and at same time frequently suggest (in this thread and others) that we should make powerplays more impactful, and so give them more power to decide games. I don't want to be watching games where the highlight of the night is the ref raising his arm. Powerplays are the worst thing about hockey (well, maybe after brain damage).
As for OP's suggestion, it makes a certain kind of sense, but it's also hard to justify, and too trivial to actually put effort into changing it.