I am reading much of your premise to be related to checking accounts (check writing and automated bill pay). Most people don't carry a large balance there so a negative interest rate on those accounts will not really have a large impact.
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.
Since most people don't do this already despite losing money to inflation, I doubt there will be a significant change in behavior for most people when interest rates tip negative.
Don't underestimate perception. Fees/taxes/"negative savings interest" are much more visible than inflation. I'd be surprised if that didn't cause bank runs.
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.
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.