I added a node-product.tpl.php file to my theme and it is completely ignored.
I reset the caches, multiple times, and tested with devel + devel_themer to see that the name was correct and it is. It says it searches for:
Why would the core theme() function not find that template?
My only answer would be that somehow the theme path is not yet available in the theme() function at the time the product is being generated.