On the plus side, requiring a reservation fee might solve some of the "no show" problems. Shelter space is often reserved well in advance, then due to any number of reasons, (often because the weather is bad), those people don't show but don't cancel the reservation. So, as far as the park service knows, the site is full, but in reality hardly anyone is there. Meanwhile "walk-ins" are turned away. I experianced this first hand last year as a walk-in. Since there is currently no penelty for making reservations and then breaking them, one can cover all possible bets with no down side.
The problem is that any funds collected will likely not stay in the park, but go into some goverment general fund black hole and nothing will really change or get better. Just like what happened with the Forest Service parking lot fees. That money quickly got diverted away from the Forest Service.