Biden is acting swiftly to make sure depositors will not be hurt, while Fed chair Yellen vows no big bailouts as usual for these banks or those running them into the ground.

Making sure the depositors will not be hurt, including accounts over the FDIC guaranteed $250,000,
is a big bailout as usual for these banks and those running them into the ground. It's like those making government decisions never took Econ 101. "People respond to incentives. That is the whole of economics. The rest is commentary."
No, it's not a bailout. A bailout is giving financial institutions to a failing business to save it from collapse. SVB has collapsed. It's shareholders and owners lost everything.
Risk management is a chess game, and you aren't thinking several moves ahead about it. SVB took excessively risky bets, lost them, and the shareholders and owners lost what they'd gambled, but they didn't lose everything. They didn't lose their skill sets and they didn't lose their reputations, because...
The insurance fund is making the bank's customers whole.
So there's nothing -- no vocal and highly visible community of angry former customers who lost their shirts -- to stop the shareholders and bank directors from doing it all over again, setting up a new bank, taking excessively risky bets, and maybe next time making out like a bandit. Average returns go up with risk, so risky bets enable you to offer depositors higher returns. When that risk isn't passed on to the depositors -- the people who financed those risky bets -- the depositors have no incentive to accept the more moderate returns offered by more prudent banks. How are more prudent banks supposed to compete with risk-embracing banks like SVB?
It's like if you have a million dollar house and buy an insurance policy with a cut-rate company that doesn't charge actuarially sound premiums. If few houses burn down they get rich and you got a good deal. If a lot of houses burn down they go broke, but what happens to you? Do you get the quarter of a million the company has assets to pay you? Or does the government swoop in and make you whole? If the government is going to swoop in and makes you whole, what's your incentive to buy insurance from a prudent insurance company? What's to stop the people who went broke just setting up another cut-rate company and enticing all the customers away from prudent insurance companies? Next year there are few fires so they're rich again; meanwhile the other insurance companies are switching to the cut-rate model too in order to survive in the new environment where they have to compete with somebody who's getting a de facto government subsidy; and the year after that there are a lot of fires and all the companies fold.
You can't blame the customers for not knowing the risks that SVB was taking. One of the top auditing firms in the country gave SVB a clean bill of health 2 weeks before the collapse!
And derivatives made of junk bonds were rated Triple-A before the 2008 collapse. People don't understand the difference between correlated and uncorrelated risk. But customers should know not to expect a million dollars worth of insurance from paying the premiums on a $250,000 policy.
If Biden hadn't authorized this, SVB's customers, large tech companies, would not have been able to pay their vendors or pay their workers. It would have been chaos.
And the same argument was made for the 2008 bailout. And that's fine: we can weigh the merits of taking the pain now versus creating expectations and incentives that make future pain more likely; maybe this was the lesser evil. But deciding Biden made the right call doesn't change the fact that this is yet another bailout.