"I think you are missing the point"

Nope, you specifically complained about having more flexible days by allowing people to carry over sick time. Now presumably you think that it's better to have unlimited sick time, but at no point have you ever actually said that. All you have done is whine about carrying over sick time.

"A fixed quantity of sick days will ensure that people show up sick for work when they run out"

Of course sick time is necessarily equal to or less than total employment time (eventually you will have to work at some point, so clearly any set number of sick time can theoretically be insufficient). Now having 40 hrs of sick time each year is by your own admission plenty for you and plenty for most people. If companies were actually losing appreciable amounts of money on their sick time policy (like you claim ) they would change it. It's easy to see that "Flu costs 12 billion $/pa" and forget that the US economy is larger by a factor of 1000 ( so less than .1 percent economic loss), as well as workplace transmission only comprising a fraction of that.

"And it doesn't stop sickness"

Of course it doesn't. Much of sick time isn't used for communicable health issues and people tend to contract communicable diseases elsewhere anyway.

"Call it paid time off or holiday"

You literally have no idea what this discussion is about. You whined about carrying over sick days and how it "doesn't make sense" (because you're an idiot), and I pointed out that fiscally sick time is identical to vacation time, so if it's okay to carry over vacation time then why is it not permissible for sick time?

"Don't pretend that this is a better system"

A better system than what? Fixed amount of sick time each year? Because that's the point of comparison. I can't compare it to whatever you are advocating for because you flat out refuse to say it. (Again I strongly suspect you want indefinite sick time, but despite having multiple opportunities to elaborate you have failed to do so).

"By having a fixed account ... {bunch of irrelevant nonsense}"

You realise the distinction you need to make is not in fixed days, compared to zero days. But fixed days compared to carried over days. If you are going to try to make a fiscal argument (again) actually try to understand what you are talking about.

Edit: You did misuse "liability", or are at least fiscally inept. You claimed that carrying over sick time was somehow reducing liability, so either you have no idea what that word means or you don't know how basic finance works. (It actually increases financial costs because you often have to pay out sick time at a higher rate if the employee pay increases.)