In america you have to realize that getting that note would actually cost people money and time.
I feel like a "sick day pool" or sick days burning PTO is just a shitty way for companies to try to save 0.02% on payroll while simultaneously pissing off the few employees who get burned by the policy. I can honestly say if I had to use PTO for an unexpected illness and it either messed up a vacation or caused me to have some vacation be unpaid, I would leave no matter how happy I was otherwise.
In Finland, where this same system exists, each urban neighborhood has its own local clinic that in large part exists just for this reason: someone feeling ill can call early in the morning, be seen a little later that morning, and get a note that they can show their employer the next day (or whenever they recover).
No, it doesn't work like that in Poland. You can just call it in, saying "I'm sick" and then go to the doctor. Doctor's note can cover up to 3 days before the visit.
It still feels stupid every time you get past that threshold and still aren't quite healthy enough for work because the cold is just taking a bit longer this time, and you have to go see a doctor just to get the note.
Usually employers are paying for basic healthcare, so you wouldn't be paying for the doctor's visit yourself. But of course somebody's still paying, and it is a bit of a waste in cases where medical attention would not be actually needed. But it's not like you have to do that every time; you can compromise by trusting people for a couple of days before requiring that. It's all about compromise.
1. Healthcare is government-sponsored, so it costs you nothing
2. There are so called "first-contact doctors", who simply specialize in everyday things like treating common cold, and in more serious cases they would just point you to another, more specialized doctor. Getting appointment on the same or next day is quite easy.
Sick days should not be a simple pool of days off that you get to treat as equivalent to holiday days. This approach has the effect of (further) rewarding people who are lucky with their health, and incentivises not taking sick days when you should.