The opportunity here for a startup bank is to replace credit card spending in the USA. The potential earnings are huge if you can get consumers to spend on a debit card instead of a credit card (because the rake is higher).
To make a checking account capture the spending market in the USA, it needs to feel premium and focused - like the Amex app. The spending analysis needs to be great, the card needs easy control in the mobile app, and it needs to inspire confidence. It also needs to feel like a trusted, approachable brand.
Schwab makes most of their money on investments. I don't foresee them trying to give mass-market appeal to their checking account, separate from the investment account.
A case study in this field is Marcus, a savings app from Goldman Sachs. If Schwab spun out their checking account into a separate sub-brand with branding that appeals to millennials and had a spending-focused mobile app, I could see them doing really well: https://www.marcus.com/us/en/savings