Well I don't know if how you are going to store your bitcoins if you don't trust wallets. Online wallet or hardware wallet are just the same type of wallets which where you are going to save your bitcoins. And that is the safest way to grow your bitcoins by just storing it there and waiting for the price increase that is going to come.
He probably meant he don't trust online wallets because they might scam you or might get hacked as well and he just believe that storing the coins in a local computer is safe, and he is right to some extent, someone should always split his coins in various wallets so that if one wallet gets stolen or scammed the coins in other should be safe.
I think online bitcoin wallet is really safe, just use the popular one like blockchain and coinbase, i'm sure there are safe. Actually there is no safe way to grow bitcoins, i mean not 100% safe. Everything always have a risk.