If you can turn that paid holiday into a vacation day and take a sick day instead, that's awesome. It basically works like this:

"We planned for you to have some time to recharge by giving you holidays and vacation days. You got sick? Can't recharge while you're sick, so that's a sick day. Take a REAL vacation day later when you're well."

I can make a similar argument in the other direction. Vacation days are roughly equivalent to money, and many companies let you convert them. If you get sick on a normal day it doesn't cost you any money/vacation. Why should it cost you money/vacation to get sick on an attempted vacation day?

If anything you're already doing the company a favor to use sick days on vacation rather than any other time because everyone has already planned for you to not be there.

... because vacation is for recharging your batteries, which you don't do if you're ill?