I don't see what's irrational about that fear. Everyone is afraid that with everyone else taking money from the bank that if they delay getting their money that it will all be gone and they'll lose their savings. Yes, I know FDIC insures most individual accounts to $250,000 but many people (1) don't trust the government, (2) are afraid it will take months or years to be repaid (3) don't trust the government, (4) afraid that the US Gov or the FDIC will go broke and not everyone will be paid or (6) don't trust the government. In the US, we've all grown up with stories of the bank crash of 1929 and 1930 and how many people lost EVERYTHING so it drives our thinking.
Yes, exactly all of that is the part that is irrational.
When banks have failed since the FDIC was created, insured people have access to their money within a handful of days, not weeks or months. It's easy to look that stuff up. Even beyond the 250k legal limit, when the FDIC manages the liquidiation of a failed bank like SVB they put themselves on behalf of depositors as the highest priority creditor, above even secured creditors. That is basically how almost everyone is going to get their money back from SVB -- because they are in line ahead of bondholders.
Also, while financial failure of the US government would be a serious problem, if you are an American living in the US it's not materially worse if you withdrew your money from the bank before it failed or not. That's the principle of sovereign debt.
So yes, it's irrational panic. That isn't to say that SVP didn't have bad practices, and maybe they deserved to fail. But a bank run is basically by definition irrational panic, and that's why the FDIC was created. Banks are incredibly useful, but even the best run bank could be destroyed in a panicked bank run and frequently was before the FDIC.
OTOH if the banks actually kept a significant portion of their deposits in the bank and on hand instead of playing financial Russian Roulette with it then people would be a little more inclined to trust the US banking system. But realistically, not a single day goes by that you don't hear of another bank, credit union, investment fund, Saving and Loan or other US financial institution that was been caught with their hand in somebody else's cookie jar.
If you are suggesting that banks not lend or invest depositors money, including longer term investments that would be tremendously unhelpful, defeat the purpose of banks and more or less permanently end economic growth. If you are saying that we shouldn't let banks write the rules on risk management for banks, then I totally agree.