The blockchain itself being down sounds a bit scary to me. I don't know if that's actually possible
The blockchain is a part of every bitcoin core (if I understand the bitcoin system correctly).
In order to bring the blockchain down, you would have to bring down every computer that is running a bitcoin core.
Maybe some super-super virus could do that. But that is nothing that would be realistic.
you are correct, the blockchain = technology driving bitcoin, it's a public ledger with every transaction ever made, and is stored on the computers and servers that run bitcoin core.
blockchain.info = a (commercial) company that offers free wallets and other services.
blockchain.info (the company) seems to have problems with handling the server load. If you're worried they'll belly-flop in the future, you might want to export your private key, or (even better) use a desktop wallet.