In this case, IMO it was more than a potential problem. On a mark-to-market basis SVB was certainly undercapitalized and possibly insolvent last week. Had there been no run, over the next year, SVB would have had three choices over the next year or so:
1. Keep deposit rates near zero, in which case depositors would progressively realize that they could be making 4.5% on their money elsewhere and withdraw their funds. This ends up being exactly the same as the bank run that happened, just in slower motion.
2. Raise deposit rates, in which case, they would start losing money until they were forced to sell assets to pay depositors their interest. Those HTM assets would then be marked to market and the bank would be officially undercapitalized and taken over by the FDIC.
3. Raise capital or sell the bank. They tried raising capital unsuccessfully. Given this, it's questionable whether they would have been successful at selling the bank because it's unclear whether the bank had a positive value.