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This issue - OneNote crashing soon after starting - can be attributed to more than one reasons.
Sometimes it is a faulty Office or OneNote update which can/should be reverted.
But there is one peculiar reason (and the corresponding solution) that I was not able to find anywhere on internet when this issue showed up on my machine - explaining it below along with its fix.
OneNote 2016 (now simply called OneNote by Microsoft) shuts down about 10 seconds or so after we start it. Event viewer shows this (under Windows Logs --> Application):
What was happening here (in my case) was that, like many other applications, OneNote was trying to access the temp location - C:\Users\<username>\AppData\Local\Temp, and weirdly - without any warning - permissions got changed on this folder.
All I had to do was add Authenticated Users with Modify permissions to this folder - as shown below.
See [Solved] Windows INTERNAL ERROR - Can't create temporary directory for instructions on how to do this.