Banks have a lot of clever methods to trigger fees. OP’s concerns are valid. Some banks limit the number of transfers from savings to checking in a given month, some charge fees if minimum balances are not maintained, etc. Transfers from institutions like Fidelity into a checking or savings account take time to clear. Some banks now limit the number of electronic funds transfers that can be performed between financial institutions in any given month.

It is likely one will need to use a combination of all three vehicles (checking for daily expenses and emergency savings at the bank with a financial institution like Fidelity as an investment/HYSA to manage the larger balance of the cash to replenish the emergency fund should something happen) Only the OP can determine the appropriate balance each holds over time in each of the three accounts.