Dear respected michaelwarman1,
Good morning! Thank you for publishing in Microsoft Community. We are happy to help you.
We went through your post carefully and understood you are encountering with the OneNote opening notebooks issue after a Windows update. And also, you have tried many methods and troubleshooting ways but still it persists. In this situation, since you have done the correct step to try clicking the Open in Desktop app button when opening notebooks in OneNote for web, it still asks for login and then provides the error saying no permission. Hence, I would suggest cleaning the local credentials and check the result.
However, after my check in the community, it seems like there have some other user facing the same issue as yours right now. See:
OneNote sync failure error code:0xE00009C8 bb0ur - Microsoft Community
Hence, we are not sure if this is a server related issue especially with the recent Windows update. Though I have checked in Microsoft Service Incident where it shows no related scenarios, I will still keep checking via online and other channels. We appreciate your kind patience and below are some information that you may give them a try before any further information updated in here.
Normally, for such a scenario, we will require customers to back up the recent notes in case something hasn't been synced to cloud completely that may cause data loss. But since you have already reinstalled OneNote and also log out/in, I assume the data are all good when you checked in OneNote for web, which is good as it indicates you have done a great sync routine in the daily work with OneNote program.
So now, we have nothing to worry and let's clean the local credentials first. For reference:
Reset activation state for Microsoft 365 Apps for enterprise - Microsoft 365 Apps | Microsoft Learn (Download the OLicenseCleanup.zip file which will automatically run the actions to clear accounts cache)
Sign in issues when activating Microsoft 365 Apps - Microsoft 365 Apps | Microsoft Learn (Follow the Remove Office credentials section)

We appreciate your understanding that sometimes initial suggestions may not solve the problem very soon. However, we can work together to reduce and resolve the situation. Therefore, please provide more information below so that we can continue working.
1. If the notebooks can be opened correctly from other devices like PCs or mobile phones. This I want to make sure these notebooks are fine.
- If the issue persists after following the clean credentials actions above, may I know the result if you create a new local profile and check notebooks from there? Manage user accounts in Windows - Microsoft Support
We appreciate your kind patience and understanding that sometime the initial reply may not resolve the concern immediately or there may have some misunderstanding about your scenario, but we would like to hear the updates from your side and let's figure out the further suggestions.
We'll be here for you to wait for your updates. I appreciate your time and effort on checking these above and thank you for your time and cooperation. I hope that you are keeping safe and well!
Sincerely,
Mia | Microsoft Community Moderator