ITIL is a very good framework if places are willing to go all in.
Problem Management is basically the next step along from Incident Management. i.e known issues that keep cropping up, problem management is to remedy that.
If you like solving long term technical issues, or investigating them, you will have a blast in that role.
Major Incident isn't fun whatsoever though. It's basically whipping everyone to get a resolution to whatever your organisation deems to be "Major". Usually hassling 3rd parties to get off their arses and if you have SLAs in place, the pressure can be ridiculous.
You should kick the security permissions on the file server over to Change Management. There isn't a Problem, it's working as designed. Just designed very badly.
That sounds like the best course to me, it doesn't seem from they way my organisation uses ITIL that it's a problem, as it is known whats wrong and it's known how to fix, I have generally only seen problems used when the root cause isn't known, or that there is a workaround but the issue is still there, changing permissions would be done through a change request from everything I have seen.