I've been using blockchain.info since then but the only thing that I had is the mnemonic words and wallet identifier.
But in the main time, I might consider moving to desktop wallet as you suggested cause I have been using web wallet since I started out in bitcoin and haven't received any bad news and I felt web wallet is less stressful for easy transaction.
You can still use web wallet as you wish but you better try out a desktop wallet as it is one of the best wallets of all. You have the private keys where you can recover your wallet anytime and anywhere you want as you wish.
I don't get it why you are feeling less stressful with web wallet as it is the same with desktop wallets.