Current wisdom was that higher interest rates would be good for a bank (after all, deposits are sticky). That didn't turn out to be the case for SVB (high concentration of risk, ...) but is more true for a more diversified bank, especially with a lifeline like the BTFP.

More importantly, the actions of the FED after SVB signaled they'll do whatever it takes to keep the system stable.

If a single bank was getting close to your hypothetical scenario, you'd probably see a forced merger. If the majority was at risk, you'd probably see actions to ensure higher margins (such as using reg Q to impose maximum rates on deposits)