I think that advice was meant for different audiences. If you're a W2 employee who has modest savings, you're in the class of people who can safely choose not to add/remove/change any of your banking relationships. You are who the FDIC was designed to protect. If you're feeling like you must "do something" then open another checking account somewhere and keep a month's expenses there. That redundancy is probably worthwhile in good times and bad.

If you're an owner and you need over 250k to be liquid on daily to weekly timescales, but hiring a three people to manage manage your bank accounts sounds unaffordable, you should find a 3rd party that can do that "cash sweep" thing for you.