"It shows I'm considerate"
Here's the belief with the best evidence behind it, and I want to give it its due before I take it somewhere else.
Alison Wood Brooks, Hengchen Dai and Maurice Schweitzer, in 2013, studied what they called superfluous apologies, expressions of regret for things the speaker plainly didn't cause. The rain. The traffic. Across four studies, a superfluous apology increased the listener's trust in the speaker, and the mechanism was empathic concern. Saying "I'm sorry about the rain" told the other person that you'd noticed their situation and minded about it, and they trusted you more for it. In one study a stranger at a station asking to borrow a phone was more likely to be lent it after apologising for the weather.
So the weather sorry works. Once, from a stranger, it does exactly what you intend. It shows you've noticed the other person's day, and they warm to you. I'm not going to pretend otherwise, and if the day's sorries were one weather sorry to one stranger, this would be a short book.
Now look at what the study didn't test. It didn't test the fortieth apology of the day to the same colleague. It didn't test the sorry before every question, to people who hear it every time you speak. It tested a single act of noticing, between people who didn't know each other, and found that a single act of noticing is trusted. The same words, said every hour to people who've heard them a thousand times, aren't an act of noticing. They're a tic, and a tic doesn't carry empathic concern, because everyone knows it isn't about them. It's about you.
Chaudhry and Loewenstein's trade from chapter 3 is what happens when the stranger becomes a colleague. The first sorry buys warmth. The thousandth has bought all the warmth there is to buy and is still spending competence, and what's left is the read.
So keep the weather sorry for strangers. It's a kind thing and the research says it lands. What the book is about is the forty a day to people who know you, where the kindness has long since been received and the trade is all that's left running.