LGIPs ability to meet redemptions depends entirely on the market, not the U.S. government. LGIPs hold very little cash and sell securities to generate cash. No one has any obligation to purchase anything the LGIP may be selling, especially during times of stress.
FDIC Insured Deposits are guaranteed by the full-faith of the U.S. Government - this is why most institutions keep some money in demand deposits at a bank. It is a simple, reliable and proven source of cash in all market conditions.