From what I have seen, they were not insolvent before the bank run. They were in trouble from a bad bet on interest rates, but were attempting to restructure their holdings to get liquidity.

Bank runs are a psychological construct, not an economic one. They are based on rumors, not facts. Deposit insurance is intended (and usually successful) to prevent runs by ensuring that depositors will be made whole even if the institution does become insolvent, because a run will destroy any banking institution.

The failure of SVB was caused by those who rushed to get their money out based on fear.