I mean, every reason you said is just reasons that management should have extreme ire directed towards them. You don't set the staff, but they DO, and they are CHOOSING to understaff you.

It is harm THEY are causing, not you.

It's still the same problem, though. Your workplace should be acutely aware humans become unwell. And plan sufficiently. I think that guilt is a relied on in any workplace, it's going to feel very much more dire in yours. I feel for you. In a perfect world, there would be enough staff that staying home sick, wouldn't matter. They could take previous years data and project how much coverage they need, and employ more (that's ignoring shortages, but that's another story to unpack). Or job share with someone, half the work, same amount of pay, holidays and sick days now don't matter, take them when you want. No workplaces need to be like this, even the shortage of available staff is a fixable thing, it just lowers profit and puts the balance and focus back on enjoyment of life. Money is the root of all evil.

Unfair of the company to provide this service without adequate staff. If the people requiring the service suffer because of short staff, they are suffering because of management, not the workers.

I really hate how management uses their failures to turn the screws on their employees that they often don't pay enough.