Unpopular opinion here: this is true for some jobs but the more responsibility you develop in your career the more unplanned absences will be noticed.
Yes, you "deserve" sick days as well as all the other supports employees are entitled to in a modern well managed organisation, but quite obviously if you do important stuff then on some days you just won't be around to do those important things.
Like this guy Jason Call for example, I presume that if he's elected is supposed to show up to congress and debate or vote on new legislation. How would you feel if he just didn't do that because he was... you know... off sick? What if he was genuinely unwell a few times when important legislation was passed and then another occasion comes around and he feels like he's coming down with something? He should just be able to take the day because his manager should have just hired more people right? No. Sometimes you've just got to grind it out.
There will be times in most professional jobs when taking a day off because you're unwell is going to let the team down. In my experience, you tell the team what's going on and see what can be done to reduce what's required of you. For example if you're meeting with an important client or whatever then maybe others can help with the prep and you show up late and leave early.