It would be great if VCs would provide simple cash management services to their fledgling investments, though.
It was dirt simple to just use IntraFi. You just say you want it, and sign the form. If you want a little more interest income, you do some cash flow planning and check the box for laddering.
Thus, an easy prudent solution is to have a second bank, likely a "too big to fail bank", with emergency funds to cover the day-to-day until you can roll over to money locked into time-based investments or the system has worked its course on the first bank.
In my personal finances, I have done that in housing deals going above the insured limit by immediately moving money into government-based financial instruments. Then as appropriate, I transferred that to my preferred investments and risk profile. If I, as a layman doing a once-in-a-decade housing deal, can manage it, then a startup can.
Its small, but also easy to at least partially mitigate.
If you still think the risk is too high, pay the premium for CDARs or insurance.
It is not about removing all risks. It is about making it acceptable. If your primary bank fails, you can manage it like any other strategic business risk.