No the typical casual user will use a lite node. Also I would point out that in e-wallets like blockchain.info your private keys ARE under your direct control. Blockchain.info only handles the higher level processing (validating tx, calculating balances, submitting tx to the network, parsing blocks for incoming tx, etc).
If the human race keeps doubling every 15 years then in a thousand or so years every square inch of our planet will be covered with humans. Assuming exponential growth will continue is flawed.
The great global Bitcoin influenza, Bitcoin Nuclear War and BItcoin famine, will keep the blockchain in check. No worries.