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>
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.
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.