What businesses want is a way to be able to track progress and communicate priority.
The difficulty for the the business is how they communicate their priority to you, that's what the priority field is for, it's a way for the person raising the issue to supply a way to communicate how important an issue is to you.
The position in the queue is how you (and presumably your team) should decide what order to tackle the work in.
The final point is the solution to the problem is to create another system for issues, which goes against the point that says "every team must have a single ordered queue of tasks".
I find it interesting that the solution to, in their own words, "One of my life’s traumas" is to add another issue tracking tool.
Are we trying to fix the right problem?