That's exactly what "sweeping" is for, in Armory. Copy the key into Armory, select "sweep" and it will search the blockchain for the balance and transfer it to the selected wallet. The private key is not saved (because if you are sweeping, it is assumed you don't trust it and shouldn't put it in your wallet).
Why would you ever want to delete a private key? If ever accidentally anybody reuses that key, it should repeat the sweeping if anything. Showing it in my address book is another thing.
This thread is full of discussion about this very topic (mainly starting on page 4).
The gist of it is that you have wallets and keys in your wallet. If you import an insecure key to your wallet, someone can "pay you" by sending funds to it and making you believe you've received money, then yank it out from under you. It's a very easy attack and completely avoidable. This is why I plan not to allow importing of keys in "Standard/Beginner" mode in Armory.
The solution would be to have a separate portion of the program devoted to maintaining addresses for sweeping. Well, sure. You could do that. But the vast majority of the time you receive an insecure key, it's because it's intended to be used just once. Ever. I don't really feel like designing and interface around something that should basically never happen, and just confuse users with the extra functionality.
That's not to say it can't be done. I just won't be doing it myself.
However, if this thread is talking about multiple people eventually seeing the same key, then it makes sense for one user to keep the key around around, monitoring the network for when it receives more funds and sweeping it right away. In fact, if this is done, I'll be setting up a daemon to do just that... Might make the whole thing kind of useless. But maybe I don't understand the application.