No. The product manager has to understand the big picture, but when you're working on a team that big, it follows that you're going to be working on a
product big enough that no one person is going to be able to keep every single small detail in their mind at once either.
You wouldn't expect the engineering manager to micromanage every single code decision—their job is to delegate effectively so that the right people are working on the right problems, and set up the right feedback loops so that engineers can feel the consequences of their decisions, good or bad. In the same way, you can't expect the product manager to be micromanaging every single aspect of the product experience—their job is to delegate effectively so that the right people are working on the most important problems, but there are going to be a million and one small product decisions that engineers are going to have to have the right tools to be able to make autonomously. Plus, you're never going to arrive at a good engineering design unless you understand the constraints for yourself intuitively—product development requires a collaborative back and forth with engineering, and if you silo product knowledge into a single role, then you lose the ability to push back constructively to make features simpler in places where it would be a win/win for both engineering and product. This is what OP means when they say that "The engineer who truly understands the problem often finds that the elegant solution is simpler than anyone expected".