Being an owner of something is a lie. Designers shouldn't feel like they own a solution, a product, a process, or nothing. They should rather see themselves as conductors. Solving a problem is moving from point A - where you have little to a considerable understanding of the problem, its implications, and desired outcomes once solved - to point B - where you have a clear vision of the solution, trade-offs, and possible future iterations. For this flow to occur, multiple stones in the road need to be moved. Designers should feel responsible for making all the necessary moves needed to happen, in time. How exactly these moves will be and which ones are fundamental results from collaborative force between designers and the rest of the team. But, designers should be empowered to think of them as orchestrators of it all.
You can understand people in multiple ways. You could rely on statistics to understand broad implications, and you could rely on conversation to understand the depth. Similarly, you can solve a problem in various forms. You could go to tested approaches and safe options or go with novelty and with your heart. Designers should be free to balance and choose between these methods. Designers should also always be inclined to search for evidence and iterate on convincing and solid ways to articulate every decision.
When designers are empowered to oversee the whole product development stack, they can make more creative and agile decisions compared to a highly specialized environment. When you see the whole image instead of just a frame of it, you make better connections and think holistically. One or two interaction channels do not define user experience, it's a culmination of all the touchpoints had with the product and company. The designer concerned about the various variables of experience is primed to excel.