Blockchain.info offers you the option to export (backup) your private keys.
How do I do that? I have scrutinised the Blockchain site and I cannot see any option to 'Export Private Keys'
Can anyone with a Blockchain account tell me how this is done?
Then maybe you are not looking well enough. You can export your private keys from their site to ensure that you own your bitcoins and only you have the authority/control over it. Also as for blockchain.info getting buggy, they are notoriously known for it, and because of that same reason I've stopped using their site for my hot wallets since the risk isn't worth it.
Correction for the two higher rank account above, there's no feature of exporting private keys ob blockchain.info except if the address was imported one and its obvious that OP was statjng the HD wallet.
But there are ways to do that, but you have to use some program, and I suggest that do it offline, here's the link,
how to export private keys on blockchain.info.
And for your concerns OP, you should have the copy of 12-word seed of your blockchain wallet, where you can use that on some compatible wallet like mycelium for mobile and electrum for desktop. Even the blockchain site becomes offline, you can still use your wallet and your funds are safe once you have backup/copy of your blockchain's12-word seed.