Hi there John, and welcome!
You've definitely come to the right place! As you've said, an adult with the recommended hours of sleep (8 hours) has about 4-6 dreams per night. Your method is quite good. What you done in your childhood years is place a mental block using auto suggestion to will yourself into not remembering your dreams. What you want to do now, is the opposite! Funnily enough.
Each night before bed, repeat something like "I will remember my dreams upon waking up" either out loud or in your head. Each are as efficient as the other.
Your method of setting a 10 minute snooze is also good. We mostly recommend people to; upon waking, try not to move too much. Close your eyes- don't think about what lays ahead in the day. Just empty your mind as if you were going back to sleep. This puts the brain back into somewhat "back to bed; dream" mode.
As for the supplements, I'm not sure. I don't have much experience with them, sorry.

Hope this helps, and welcome again.