Question and background:
Given that each institution's accounts are FDIC insurance limited to $250,000, is there a service which helps someone invest, say, $5 million, with 20 different banks, so that each is covered with the $250,000 FDIC insurance? That service would ideally coordinate and simplify the 20 different accounts which would be set up.
Yes, you'll be looking for a service called "Liquid Insured Deposits" or similar name. Various broker-dealers and financial firms offers such a service, though it may be limited to, say, $2.5 million (10 banks) instead of $5 million.
Essentially, you deposit your funds into the program. The service itself then pools your funds with deposits from other investors and allocates the funds across several banking institutions such that no investor pierces the $250,000 FDIC threshold at any one bank.
Hope that helps. All the best.