Troubleshooter or troubleshooting has nothing do to with shooting anything. It is the art of problem-solving, and the term originated in the late 18th century or early 19th century when technicians were dispatched to find problems in telegraph and phone line infrastructure, and then find a way to repair or work around the issues. Historically then, the term troubleshooter meant to find and solve technical problems and issues.
Today, troubleshooting can mean any effort to analyze not only issues or problems with technical systems, but also is a methodology to find and address issues within any organizational or political situation, and then recommend how the situation or event can be repaired or resolved.
Good troubleshooters will analyze the situation, and then make an educated guess on the most likely cause of the problems being encountered (based upon logic or previous experience), and focus on those potential problem areas first. Once the problem is identified, they will attempt to isolate and fixed the issue as quicky and efficiently as possible.