Since both accounts are covered by FDIC insurance [0], it seems that the only difference (unless I am missing something) lies in the rate of return, thus ceteris paribus and the checking accounts have a better rate of return compared to savings accounts. Both have real negative rates, so the situation would be an active discouragement to hold the government-managed currency in any form.
Regarding cash holdings, you are right that they are probably a bad idea. However, if numerous banks could potentially experience a crisis beyond the intervention capabilities of the Fed and Federal government, then it would probably be more sensible to move the money into cash and store it in safety deposit boxes, possibly across multiple banks. This to me seems an unlikely scenario, but not necessarily impossible.
Yet on savings accounts, or those accounts whose intent it is to be liquid yet remain relatively unused, could be impacted by this. Which at that point, again under your premise of robbery, a person is essentially wagering whether they would rather be guaranteed to be fleeced incrementally, or run the possibly of someday being taken for all of their savings. At that point I think you would see many people start to withdraw large amounts of savings, whether to be stored underneath a mattress or into bonds or some other investment, while still keeping an active checking account.
For my part, I already carefully manage how much I keep liquid versus invested, but anything equivalent to a fee would cause me to switch institutions or otherwise drastically change this strategy.