"It's not how I was brought up"
A lot of readers will recognise the habit from a house. Nobody in your family talked about things. Worries were carried, quietly, and you found out about the big ones afterwards, if at all. Your father's health, your mother's money, the trouble with your brother. All handled, all silent, and the silence was what a grown-up looked like.
I'd separate two things. What the house taught you about worries, and whether it was right.
What it taught you was that a worry is a private matter, that telling is weakness or imposition, and that capable adults handle their own. It taught you by demonstration, over years, and you learned it well, because children learn what's demonstrated. That's not a character. It's a training, and trainings can be looked at.
Whether it was right, you already know, because you were on the other side of it. You found out about your father's health afterwards. How did that feel? Most people, asked this, don't describe relief at having been spared. They describe hurt, and something like exclusion. The silence that was meant as protection arrived as distance. You know, from the receiving end, that the house's method didn't do what it was supposed to, because you were the person it was supposed to be protecting and it didn't feel like protection.
That's the most useful thing about having grown up in such a house. You have direct evidence about what carrying it alone does to the people it's carried from. You've felt it. The question this chapter asks is whether you want to be the parent, or the partner, or the friend, whose worry the people around you find out about afterwards, and feel about the way you felt.
The house isn't destiny. A habit learned by demonstration can be changed by a different demonstration, and the first person to tell in a silent family is usually surprised by how much everyone else had been wanting to. Silence in families is rarely unanimous. It's usually a thing that everyone is doing because everyone else is, and one telling shows the others that it can be done.