The people who do follow the rules are left feeling shorted by those that don't. That doesn't seem right.
I think this happens with a lot of things. The employer can say "be trustworthy" and save money across the entire pool of employees, but the employees who follow the rules have to stand by watching other who break them.
Same thing happened with marking spots in my apartment complex. They numbered the spots to go with apartments, but don't enforce it. Someone always takes my spot. According to their rules I'm supposed to park further away in the unmarked spots, or steal someone elses spot. Most people follow the rules, so the office gets less parking complaints. But now people whose spots are taken are left worse off than before.
How is that better than just giving everyone the same number of days to manager as sick or personal for themselves?
It's also a matter of solidarity that those who can support those who can't. In this case, the healthier employees support the ones who are ill.
According to your logic above, why don't we get 100 sick days per year instead of 10? Some people need that many, fellow employees should support them too, right?
I guess the sick vs vacation day thing becomes a more of a problem in an environment where the rule-followers see a lot rule-breakers taking advantage.
Yes, all of these can be checked to some extent, but it will not be possible to fully check everything.