The process you have now is simple, effective and doesn't cost much.
I've seen attempts to put this into a workflow solution and it does add overhead plus someone to manage the workflow solution. Sometimes you are not going to get there if the payback isn't there.
That is, if it increases the time it takes to get things done, why change this? Yes you write about tracing the work, but is there a gain in productivity to offset the costs? This is unlikely.
The method you have now is cheap so selling the company into a system will be a rough sell. To make this happen it has to happen at the top. The owners have to direct such to get done.
https://en.wikipedia.org/wiki/Workflow_management_system for example.
We are a company with ~250 employees and do back office work for clients. It is similar to accounting but different. We get hundreds of pages of documents from each of our clients daily and they are spread over a few files. Each file having multiple pages. Someone picks up the files from he FTP'ish folder and then assigns them to different people to work on them. Most of the work currently is being done by just copy pasting into different folders which are called "holding tanks", "in progress", "done" etc, renaming files and so on. As you can imagine there are many places where things can be missed and errors made. I am trying either develop or purchase a system to get away from these folders and make this process more traceable. Would appreciate any ideas. We have created a simple "task" system where someone can make a task, attach a file and assign it to someone else who can update status, add notes etc. It can work but I am getting a lot of push back because it is not as simple as cutting/pasting and doing this hundreds of time in a day can lead to overhead.