French banks are actually doing that! SG/Société Générale, a brick and mortar bank that costs a hundred dollars a year, built Boursorama, an e-bank that's free and that gives you $200 if you subscribe. Advantage: SG has too many manned agencies and can't justify preemptively mass-firing them yet – they first need the customer base to dwindle. Better IT, usage of Internet techniques at the core, and drastically simplifying the product line is an intended consequence.
Same happens with other French banks. As a personal PoV, banks' manned agencies provide bad service (from 20x delays to plain mistakes), so I won't cry for them.
Boursorama is not a bank. It's a website to allow people to play on the financial markets ("bourse" = exchange). I think Americans would call that a sort of dealing account.
Obviously, a dealing account has to hold funds and handle transfers. That's not any close to a consumer bank though.
Boursorama Banque is a bank, as in, a normal cheque account, IBAN, VISA card, loans, term deposits and no trading account. Real normal bank. The main Boursorama website is another department, it's not really clear on their website.
Same happens with other French banks. As a personal PoV, banks' manned agencies provide bad service (from 20x delays to plain mistakes), so I won't cry for them.