It's not so much what any one bank does, in isolation, it's the incentives it creates (or distorts) for depositors. One obvious reason we have limited deposit insurance is the insurance fund is not big enough to backstop all deposits. But another critical reason is we want wealthy, non-Mom and Pops to have some skin in the game with the loans (deposits) they are making to banks. They have the sophistication and financial wherewithal to monitor bank risk taking and move their funds to safer/higher-quality institutions since they have risk of loss if things go south. This type of monitoring is one of three pillars the FDIC views as playing important roles in mitigating the moral hazard problem with deposit insurance.
This moral hazard problem arises with any insurance, but it can be exacerbated by government distortions. When the government, via flood insurance, bails rich people out of their foolish decision to build houses right on the coast where there's - wait for it - lots of water that is bad for houses, it encourages more foolish decisions to build houses right on the coast. SVB was reportedly paying 5% plus on large uninsured deposits. We want the VCs and tech companies to pocket the gains from this but then socialize all their losses? Ok, but then you'll get more SVBs and flooded houses on the coast that we all pay for. Another approach is to let the uninsured depositors learn the tough lesson of making risky on-demand loans to liquidity-strained borrowers.