Go ahead, try and pull your money out
Banks use US bonds to run their business model. It is like the engine block to a car. It is how they pay interest on money held at the bank and how they balance assets and liabilities.
However, when interest rates on US bonds are raised suddenly it means the bonds the banks are holding that pay a lower interest rate is now worth less. You have to sell them at a discount, or loss.
So to avoid taking losses across the board the banks categorize the bonds as something they will not sell until maturity, in that way they argue they haven't lost value, but that is truly deceitful and once you categorize them that way if you sell them you have to pay a penalty.
All banks in the US, but especially the regional ones have tried to hide the losses since interest rates have risen so much. If people take their money out of the bank then they must sell those bonds and when they do they realize a double whammy, the loss as well as the penalty.
Meanwhile these banks make money by making car loans and mortgages. However, those are drying up since interest rates have made affordability of houses drop drastically among Americans. Fewer people can afford to buy a car or house, so they are doing fewer loans and making less.
It is important to understand these structural issues, because the problem is not that people need to be reassured the problem is that the bonds the banks have lost value and the loans they make to make money have dried up.
So what Biden has done does not in any way address the real problem, meanwhile it actually makes it worse! He has basically claimed that FDIC which is designed to insure a very small portion of bank holdings will now insure all of them! How? Insurance companies assess the potential liability (which in this case is around $6 trillion) and then charges a fee to collect the money to be ready to handle those risks. If you started at 0 they could charge a small fee and over time have a large enough amount bankrolled to cover any potential liability. But that is not what Biden has done. He has said overnight, when every single bank is at risk, that the insurance must be able to cover about 8x what it was previously covering. There is no way to collect those fees. Imagine if you put your money in the bank and the interest rate was 3%. I don't mean you get 3%, I mean they take 3% each year. No one would keep the money there. That is what they will have to do. Obviously no one will want to do that, so they will have to make it very difficult to remove your money from the bank, even putting in a big fee on anyone withdrawing money.