Seeing as you don't want to go down the machine polishing route, I've had very good results by hand on very neglected black paint using
Soft 99 Scratch Clear Wax
When I first saw the ad I was like "yeah.....right!"
Then I tried it on a friend's neglected black paint using the same method as in the YouTube video just rubbing away until the polish disappeared.
Did a little video with the left door untreated.
5 mins of rubbing on the door on the right and these were the results.
Click on the below