This settles it. You don't know what a blockchain info wallet is.
They DO NOT have access to your private keys....
And you dont know what security is.
"Let's just deposit 10.000
BTC on an online site exposed to all sorts of hackers, and also take their word for it that they won't have access to it"
Basically this is your logic, its just a question of time until something will go wrong there, and it had, just look at the testimonials of people got hacked by using online wallets, many from this forum, so look around.
Blockchain.info is not some kind of scammy site who would run away given that they have received so much findings from investors. The code is 100% opensourced and people has confirmed that the private keys go to the server encrypted and they would have no chance to get it unless you tell them the password. Most of the hacks are the user's fault, using weak passwords and no two factor authentication.
Ok then even if that's true, there are keyloggers and other scrips that can read passwords, what about those?
See why would anybody risk 10.000
BTC, and expose it to unnecessary risk.
Now i say again, under perhaps 100
BTC the security could be enough, but certainly for really large sums i would definitely not expose my wallet to unnecessary risk.