"If I don't, they'll think I'm arrogant"

The fear that stops most people is a picture. You come through the doorway without the sorry, or open the email without it, or say the thing in the meeting without the "can I just", and the room registers a person who thinks a lot of themselves. Arrogant. Entitled. The one who doesn't say sorry.

Kenneth Savitsky and colleagues, in 2001, studied how people expect to be judged after small social slips compared with how observers actually judge them, and found the gap consistent and large. People expected far harsher verdicts on their own conduct than anyone gave. An earlier book in this series uses this finding for the fear of saying no, and it applies here with the same force. The arrogance you picture is the forecast from inside the moment, and forecasts from inside the moment run harsh.

Consider what people actually think when you say "thanks for waiting" instead of "sorry I'm late", or "quick one" instead of "sorry to bother you". In my observation they think nothing at all. The sentence is ordinary. It's what Dev says, and nobody has ever described Dev as arrogant for opening a question with "quick one". They think about the question, because you've given them a question and not an apology to respond to first.

Arrogance is a different thing from not apologising for the rain. Arrogance is not apologising for a fault. It's blaming the brief, walking on in the doorway when you were the one not looking, taking credit for the north report. None of that is what the method asks. What it asks is that you stop confessing to things nobody accused you of, and the person who does that isn't read as arrogant. They're read as a person who knows the difference between a doorway and a fault, which is to say as an adult.

There's a longer-term version of the fear, that repeated non-apology builds a reputation. Run it the other way. Repeated reflex sorry has built one, and it's the read from chapter 3, and you've been living in it. Whatever the risk of being thought confident, you've been paying the certainty of being thought unsure, every day, in exchange for avoiding it.