Well, I ended my subscription to Office - so technically I do not have the desktop version - However, it still works (but with persistent nagging).
Your suggestion is stupidly obvious - yet it just did not occur to me! Onenote does not need to be specifically uploaded, the web version of Office should work just fine - the stupid thing is, I have used Onenote online many, many times before.
Big thanks for kicking my brain into first gear. .