I certainly prefer to think of debugging as a process of learning about a system rather that trying to try things at random without making a mental map of where I am.
When you stop to report on your progress, you don't need to deliver detailed notes. It could be as simple as "yeah, fixed it" or "figured out the problem, I need three hours to refactor the code to resolve it" or "based on what I'm seeing, Bob would be better at fixing this; I can spend a few minutes to show him how to reproduce it."
"try things at random" is not debugging, so you're right about the process.
Sometimes my method is instead to look at data output at a certain level of abstraction like system calls (which is usually not helpful except for detecting a network hang or a path config whose directory doesn't exist) or database queries or something. That is usually more helpful if I either already suspect I know where the problem is and need to know what model name or file name to `ag` for.
If (Hephaestus be praised) there is actually an automated test framework in place, I'll start by writing a failing test or copy-pasting an existing test and modifying it to fail, then (if it isn't JavaScript), inserting a debugger into a function the test calls.
All of this tends to be pretty successful.
But I've occasionally found myself on a project where I just have a sense that I don't know what is going on or how pieces fit together and I feel a strong temptation to stop and fix that sense of bewilderment. When I resist this urge is when it feels like I am trying things at random.