If 20% of a small number and 2% of a bigger number are the same number of people in hospital, the hospitals will be in the same situation. What is emotion driving? People showing up in emergency? These people will be triaged, and will not take up the most limited resource - ventilators - unless they need them.
I don't think 'panic' will overwhelm anything other than the triage nurses. This is not what you should be worried about.
Take any disease that kills 99% of those that get it. You are going to be a lot more scared of it, than one that kills 0.1% of infected targets. It does not matter if the overall death number is the same.
People will act far less logically in the first case than the second.
This is not about "math" it is about emotional responses and how that affects everything. As I said, you are more likely to be trampled to death than die in the fire that caused the panic.
I don't care about what tugs on people's fears the most. I'm not afraid of their fear, I'm afraid of hospitals being overrun by actual cases.
You are more likely to die in the stampede triggered by a fire in a crowded cinema, than the fire itself.