Take a look at actual online wallets, such as blockchain.info for instance. Is that possible to create the same type of online website but that allows cold storage wallets creation ?
Yes, there are already services that
claim to be doing it. Xapo's vault is one example.
I don't like to tell others what to do with their money, but when creation of a paper wallet is so cheap and is so much safer, why would anyone believe online services'
claim, instead of doing real cold, offline (paper) wallet?
Normally even the exchanges should keep a big chunk of their assets in cold storage, since they don't need all the bitcoin (the trades are off chain). But the Bitfinex hack shows that this doesn't actually happen as we'd think.