It’s more about the definition of “sick”. I’ve seen people take a week off because they had a bit of a throat/head ache. And others that worked from home when they had a flu.
Honestly? I stay home and might check email when I'm bored but don't do any real work, but I don't usually file sick days either because the bureaucracy is a pain (seriously, I'm already sick, I can't deal with that added stress and I'm not going to the doctor just to spread my viruses and be told to go back home and rest). And as long as we're talking about 2 or 3 days, nobody really seems to care.
I've always had managers who care more about what I deliver than when and where my butt is in a chair. Maybe I'm lucky, maybe I'm good enough that it works out that way.
No, I mean, that’s perfectly acceptable. If your boss cares only that you deliver, and doesn’t notice/care that you are gone for a few days due to sickness, that’s still not on you.
Have you considered just informally notifying them that you're out because sick? I mean, not doing that is fine too if you can deliver, but informing them might build more trust.