It should not matter to the employer why an employee is not working.
If the issue is insufficient time off for illnesses, then the negotiation should be for more paid time off (PTO) in general.
This eliminates all the resources wasted on catching fraudulent use of sick days.
In many jobs with separate vacation and sick days, the vacation days still require the "advance notice" for scheduling, but the sick days do not require that one know, days in advance, of an unexpected illness. That gives the employee a few days that can be used for unexpected things, and the employer some assurance that a maximum of X days of unexpected outages will occur per employee.
More often than not, the "only one type of paid leave" comes attached with the "requirement for advance scheduling" of those same days, which does not provide for the unexpected illness situations.
The true cruelty of "no sick days" isn't that employees have to burn vacation days, it's that vacation days have to be scheduled in advance and approved. Neither of those things leave room for sudden illness or injury, and prioritizes company schedule needs over things like scheduling doctor's visits.
States with sick leave policies have stipulations like these:
https://www.lni.wa.gov/workers-rights/_docs/ReasonableNotice...
If you need to miss work unexpectedly, you do so. The employer just trusts your judgement that it's necessary. If it's going to be more than a day or two, you are expected to talk with your employer and explain what's up -- mostly so the employer can adjust assignments and schedules with a better understanding of what to expect.
Do some employees abuse that? Sure. But, in my experience, that's rare and the people doing it are usually poor workers anyway. They can be handled through whatever the normal review mechanisms are.
This is my most common experience. I don't know how common it actually is, and I'm sure other situations would require different, perhaps more formal, procedures. But it's worked well in the places I've worked.
Perhaps a mechanism that allows for unplanned time off, with a little education on how to properly plan PTO to allow for unplanned leave.
If the label between sick and vacation is meaningful, are "Special Inspectors" needed to ensure people are sick when taking a sick day?
California requires overtime after 8 hours of work per day and on the 7th consecutive day.
If that is not enough, make overtime cost 2x.
Lots of ways to ensure employers do not run employees hard without getting into details of sick days and whatnot.