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The paperwallet provides a seed-phrase along with the private key and a few addresses. I would be fine if I have a seed-phrase and not the private key, but I am paranoid I would be worried about something happening with the seed phrase and losing the funds because I didn't keep the private key. Then again, I do exactly the opposite. I keep private keys and don't tend to use the seed-phrase, but I keep it anyway as backup. I store it all in a text document so its not like a couple more bytes takes up needed space
I suppose it just comes down to you. How much you are storing in the address and your risk tolerance to determine if not saving the private key also is worth it.
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