"It's just politeness"
Start with the belief that sorry is free. It's a courtesy word. It doesn't mean anything. It's just what you say.
Shereen Chaudhry and George Loewenstein, in 2019, set out a theory of four things people say that seem cheap and aren't. Thanking, apologising, bragging and blaming. Their argument, tested in two experiments, was that all four carry information about credit and blame, and that each involves a trade between two things people want to appear to be, competent and warm. When you apologise, you take blame onto yourself, and that reads as warmth, because you're putting the other person's feelings above your own standing. It also reads as a small loss of competence, because a person who's at fault is a person who got something wrong. The words are cheap to say. What they do to how you're seen isn't.
Apply that to the day's sorries. Forty a day, none of them for a fault, and every one of them a small trade. Each says, to whoever hears it, "I'll take the blame here", and each one buys a little warmth and spends a little competence. Over a day that's forty small spends. Over a year in the same office it's thousands, and the people around you have been keeping the balance without knowing they were, and the balance is the read. It says warm, and it says junior, and it says the sort of person who'll take the blame, and none of that was your intention in the doorway.
So "just politeness" is true of the word and false of the effect. The word is a courtesy. The effect is a transaction, and the researchers' point is that the transaction is what makes the word worth anything. If sorry really were free, it would carry no information and a real apology would mean nothing. It's because sorry costs competence that the real sorry works, and it's for the same reason that forty reflex ones a day are expensive.
I'm reaching from a theory of communication to a doorway, and I'll say so. But the shape holds. You've been treating a word that has a price as a word that hasn't, and paying the price all day.