It doesn't matter to your shares if the bank goes bankrupt, because the shares are stored at a central depository. And it doesn't matter to the central depository if you bought at a neobroker or your house bank. Your risk is much higher with your bank fund and the only one who makes a return is your bank with the fees.
@Der_Dividenden_Monteur as I said, I don't know how it works with online brokers, with normal house banks it's like you say. If this is also the case with online banks, I trust you.