If they can't handle more customers, they should close signups until they've staffed up. Like how a store should not accept orders they cannot fulfill.
Yeah and the boards will kick out whoever that decided to stop accepting new users when they put much effort on marketing just because they couldn't get 100% customer support.
That would never, ever happen at a VC funded tech startup no matter how bad the situation was. All investors care about is growth, and all the C-suite cares about are investors.
Do you even hear bad things about small countries? That doesn't mean they have ace governments.