We (local council with 500+ permanent employees) had the same issues (our efforts were not visible + company vs. individual view) and solved it in the same manner - using SharePoint at the time.
Broke it into "issues to resolve" and "unexpected problems"...and added "scheduled outages" as well.
Also made the list public.
And made the escalation process transparent...you talk to me, if you don't like it get your boss to talk to my boss, short business case for bumping up the priority and that filters back down again one way or the other.
And the biggie - the helpdesk...re-arranged into 2 levels of support, general (1st level) and app (2nd level).
All those phone calls to developers "just because you know them / their extension" were met by "What's your case reference?" and "Oh, don't have one? Well log it with the helpdesk and we'll get on to it." and "If it's that urgent (aka you think you're THAT important) get your boss to talk to my boss".
We realised we lost no end of time to answering daft calls or answering the same thing over & over ("yes, the server blew up, that's why your application is not working") so the new process was more cumbersome but significantly more productive overall.
And people always react better to knowing the worst than not knowing...