Ultimately the VCs that guided their portfolio companies to use SVB and neglect obvious cash management are responsible for the anxiety. I suspect they would have stepped up to fill any gaps, which would have been less serious than claimed; making payroll was never seriously threatened.
But before then they were happy to let some useful idiots clamor for a depositor bailout that would absolve them of that responsibility.
The systemic risk became real simply because a lot of people realized that SVB was not the only bank with problematic assets. The result would have been an outflow of deposits to the "too big to fail" banks because those banks effectively have infinite deposit insurance (because everyone already knew they would not be allowed to fail).
Failure of a number of regional banks does not threaten the whole system but the handful with sketchy solvency / liquidity.
Also, many other small banks are not fine. SVB is not the only one with concerning duration risk in their assets.