Somebody from Robinhood's legal team either said "yeah, the SPIC will totally be okay with this" or management ignored any advice to the contrary, whichever it was those people need to pull their heads out of their asses. Even Schwab, an investor bank, has insurance from the FDIC for their deposit (and sweep) accounts - if they could have gotten away with just SPIC coverage don't you think they would have?
As well as Fidelity's checking account, Cash Management Account, is FDIC insured. This definitely feels like a case of Robinhood ignoring some advice.
However, Fidelity's trick is to sweep all cash into third-party FDIC-insured bank accounts behind the scenes. This yields several benefits:
1) Practically speaking, it offers the exact same FDIC insurance as a real bank account, because your money is being held in a real bank account.
2) Since cash is deposited into FDIC-insured accounts behind the scenes, Fidelity can seamlessly split money up so that each sub-account holds no more than $250k (the limit of FDIC coverage). This is how they can offer $1.25M of FDIC insurance (split between 5 accounts).
The customer never deals with this complexity, as it's completely abstracted away, behind the scenes. The result is a cash management account that offers $1.25M FDIC insurance on cash balances.