@JillCyCian Hi. With a long work history, I think it might be better to try the skilled visa route first; student visa is expensive kahit vocational pa ang kukunin mo. With a relatively big family, mas mahihirapan ka na i-support ang family on your part-time income as a student (unless your wife will work full-time and leave the kids to childcare, which is expensive too)--if they're going with you to AU ha.
You can nominate an occupation that is related to your work experience, and then study automotive once you get a PR visa. From what I know, the tuition fees for PRs are lower than for international students, so it might be worth waiting a bit to save money in the long run.
As a student kasi, you are limited to 20 hours per week, and it's not easy to find a job that might be related to your course (if you're trying to build your local experience). Most people I've asked said that working part-time to pay for living costs is doable, but it will be difficult if you are also working to send money back home or if you're supporting your family in AU. Once you graduate from a 2-year course (whether vocational/higher education), you might be eligible for 485 temporary graduate visa, which allows you full work rights depending on the program you've completed. Mind you, this is NOT a PR visa. You'll have to lodge a separate skilled visa application to be able to migrate your whole family to Australia.
As to your question regarding courses regarding student visa, you can enroll for vocational courses (like automotive) as long as the school and the program are CRICOS-registered. If you mean whether you can use that vocational course to migrate to Australia, I believe it's possible, but you have to choose whether to nominate your oil-and-gas occupation (which is based on your current work experience) or automotive-related occupation (based on your Australian studies).
GTE is unrelated to the process of getting full-time work. It is a statement to certify that your plans to go to Australia are really for study purposes; DHA (immigration) uses it to "measure" how legit are your plans to study in Australia. Of course, plans can change along the way and during studies, so if the student decides to stay in Australia to work full-time, they can do so as long as they have the proper visa (485, 457, 189, 190, 489, etc). Typically in the GTE, you emphasize your intentions to study lang. While I think it's possible to mention your future plans about working in AU after your studies, I would advise against it because you don't want to give DHA a reason to outright reject your visa application (because it will look like you do plan to stay in Australia beyond your studies).