You're going to get users who sign up thinking this is just like a bank account, but with better returns. Then the market will drop 10% when their rent is due tomorrow. Bank rates are so low because the money is always there and insured. Anything without these features should not be called a checking account.
You call yourself Bogle fans, but passive index investing and instant cash access are fundamentally opposed from a time perspective. Finally, this is a tax nightmare. People will think they have huge gains in their account but get hit with a capital gains tax when they go to withdraw.