You're one natural disaster away from being completely fucked.
Imagine a regional power outage lasting a few days. Now imagine it lasting a few weeks. Phone and internet out. No one will be taking your cards. Cash only and no functioning ATMs for miles. That could never happen, right? Except for Hurricanes, earthquakes, tornadoes, ice storms...
For this stupid cash delivery service to be useful in a disaster, they would need to: somehow survive the disaster, have access to a source of cash* that other people in the region don't, have drivers willing to deliver it in a disaster zone, and finally NOT dip into the cash themselves to help their own families.
*Oh and to protect that "source of cash" they would also need premises and a security team.
If you're relying on a "cash delivery app" to bring your cash during a disaster, you're already screwed.