Buying it is one way to do it and I have noticed the same thing. I am mad that I can't run Amazon giveaways on my shirts, a lot of shirts I can but none of my own for some reason. That would be the magic trick :-)
Sending them to influencers is a great idea.
There's a workaround to run Amazon PPC involving AMS and Amazon Vendor Express but its a pain in the butt to setup, I'm in the process of setting mine up right now.
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I have never got AMS to be profitable, not on shirts or my private label products.
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