Yep, you can structure the roles however you like. I'd guess every plausible combination has been tried.
In my world engineers are first class citizens who have real input to the product decisions and meet customers, the PMs are usually technically capable, the PM job is too big to be done part time and hence isn't. It is what it is, and doesn't seem to be close to how things work in your world. But in my world, the product job is more complex than "someone has a need and you give them a solution". For internal development or consulting, that's probably right.
In my world engineers are first class citizens who have real input to the product decisions and meet customers, the PMs are usually technically capable, the PM job is too big to be done part time and hence isn't. It is what it is, and doesn't seem to be close to how things work in your world. But in my world, the product job is more complex than "someone has a need and you give them a solution". For internal development or consulting, that's probably right.