To the best of my understanding you can’t initiate payments with a money market fund using, say direct debits or visa/Mastercard. So they can’t replace an ordinary bank account.
Can't replace it. But with some trivial API glue and a cron job, you can get away with storing next to nothing in the bank account on most days as long as all of your credits and debits are scheduled and upper-bounded. E.g., if the debit side of your entire financial life is payroll + rent + credit card bills.
But this shifts risk from your FDIC insurance to your SIPC insurance.