The railroads hit the core of the problem. No employer really objects to sick employees staying home. The problem is when that is abused into just becoming time off whenever the employee wants a break, often on Fridays or Mondays. And for a time-critical business like railroads, that just doesn’t work.
Perhaps Mondays and Fridays not paid sick days, but other days would be.
If you are sick, you are not out spending money.
Perhaps cap sick pay at $100/day.
As Dilbert’s Pointy-haired boss pointed out, 40% of sick days are taken on Mondays and Fridays.