|The right way is the way the client is most satisfied. Each client is different and each situation is different, so there is no universal solution. What you call fooling around for days/weeks is called troubleshooting and helps both the client and the consultant learn.|
There have been times when I've worked continuously on a single problem all day and late into the night. In the end, not only do I get great satisfaction for solving the problem, but I also learned why it happened, how better to prevent it in the future, and how I can solve it again if I or someone I know runs into it. I wouldn't gain that knowledge or personal satisfaction had I just given up and reinstalled.
Depending on the situation, your way will not only delete important files and precious memories, but it will end up being slower as all the software needs to be reinstalled, personal settings to be adjusted, and any files/systems that were backed up need to be replaced. In plenty of situations, your way is indeed slower than mine.
As you said, learn. You haven't had much success fixing a problematic system, but I have, and I continue to get better with each attempt. To each his own, and that's all I have to say on that.
Apologies if I don't respond to your reply immediately. I don't check this site daily, but you're welcome to PM me as a reminder.