data:text/html,<html contenteditable>
You could probably set that as the URL of your home page to get the behavior you described.
I add a style="height:100%" in the HTML tag so that clicking anywhere puts focus on it for editing. You could add other styling and initial content as desired.
For notes that I've refined and want to hold on to, I tend to use a physical notebook. I enjoy the kinesthetic process of writing by hand so I still use that for more permanent notes.
There are local-first file based solutions (like classic TiddlyWiki) that could work for you,.. but at that point maybe you meant to open your favorite text editor or IDE.
I kind of like that this simple solution always opens a scratch buffer and I have to put it somewhere else to save it (never saved accidentally).
It also keeps the source formatting (or with ctrl-shift-V ignores it), which can be useful aside from the note taking itself.
data:text/html,<div contenteditable style="height:100%"><script>addEventListener("beforeunload", (e) => {e.returnValue = "?";})</script>