Most of the time, I don't finish fleshing out an idea on paper. But the act of writing (or drawing) it somehow helps me explore new concepts faster. After that I just continue coding or whatever the next steps is. It's also useful when I want to sort or categorize different ideas/thoughts into sections.
I'd like to add that it's really nice to have a notebook always in front of you. Mine is always open & ready to be used. It sits right in the middle of my split keyboard so it's literally always in front of me.
What I'm saying is: a notebook and pen is not an essential must have tool for programming. If you use them then all power to you, but if you don't then do not feel like you're doing things incorrectly, and don't listen to people telling you otherwise.
For me the notebook isn't actually for memory. Those kind of notes are better on digital so I can access & edit them everywhere (I use notion for those). The only times I'd read what I scribbled is when I context switch and need to remember what I was doing. It's more of a scratch pad really. Figma also works well for this. Just depending on the work.
I will say that writing or pretending to write notes is almost a super power. In the same way that some people ask "you don't write things down?", I found that people tend to think I'm more disciplined or hard working when I'm on my notes.
Can't say I haven't pretended to write notes before :)
And, as I mentioned to my wife earlier this week, if I feel stressed or something has upset me then just the motions of making coffee or tea is relaxing.
This reminds of the character Ko Murakami in the manga World Trigger, who can master anything after trying it once and then sleeping on it. Great read if you like manga around strategies and battles.