when there is a bank run, fdic is going to default on their promises
Exactly. The concept of looking after your own money and belongings seems so foreign to people. It surprises me to see that kind of mentality here.
The FDIC is backed by the federal government and debt issued by the federal government is assumed to be essentially risk free.
As of now the FDIC's insurance fund is ample to be able to pay claims on funds lost by failed banks. Their insurance funds is receiving more in premiums now then they are paying out in claims.