Although I appreciate your concern, learning to perform the test does not ensure an inspector will allow you to do so.
I'm sure you are competent at what you do. And it sounds like you are completely frustraighted with your current service provider.
The red (orange?) wheel in your picture is the governor tension weight. Is it moving freely? It appears to be hanging well.
If you are dissatisfied with your current company, call another company or ask your inspector to recommend three other companies.
Not all companies & customers are a good fit for each other...
