In some ways, money market funds are safer than banks because they're not playing the borrow short lend long game. The risk of getting wiped out is also lower, though the risk of getting 95 cents on the dollar is higher.
If I understand you correctly, you're basically saying why use a bank versus something else? My answer would be because for vast majority of retail bankers it doesn't matter. Same reason many would use Gmail vs self-host. They want to put their money somewhere and ensure that it will be there when they need it. As a society we have a framework around that and we provide deposit insurance and we call the orgs who take your deposit, give you an IOU (in the form of an account balance) then lend a fraction of it to others a bank.
For significantly larger sums no I don't think it makes sense to use a bank and most large corps have a treasury that specifically deals with that.