I was simply going to swish (don't remember the term) with Blockchain when I decide to spend it in the future. They only require the private key correct?
I believe the term is sweep.
I've never used the online Blockchain.info wallet so I'm not sure, but I think that is correct: they only require the private key. Mt. Gox had a sweep function too, and many other wallets, like the Android bitcoin wallet and Electrum.
Depending on the amount you saved in cold storage, I would consider which wallet to use. In my case, I've stuck with Bitcoin Core since I started, so that's what I'm comfortable with using, despite the large amount of space needed if not using pruned mode.
I might consider using a brand new wallet with Core (the new ones are HD) and sweep all my funds to addresses there, and then prune that or something. Maybe.