Everybody likes a good us versus them story. Well, the ongoing narrative that "product" thinking is better in every way from "project" thinking isn't one of them.
There is no doubt that it was borne out of frustration, but that was related to a specific set of practices and methods. These change and evolve over time.
You need to manage both your product and projects, and you can suck at both. Or you can excel. It has nothing to do with eliminating one of them, and all with your management and leadership skills.