An airline that sold out of all its seats 6 months ahead of time would be considered incompetent, but governments do this all the time with reservations for various activities, because they are trying to "be nice" by charging low entry fees.
Increasing prices will only exacerbate the existing issues around equitable access.
Here's an article[1] describing a University of Montana study on the subject, for anyone who would like to consider the issue in greater depth.
In any event, that unlikely to be helpful in this specific situation, as the majority of all revenues from rec.gov go directly to BAH[1], rather than to the public land management agencies responsible for stewarding the resources in the public interest.
BAH also has price setting authority, with a dearth of public oversight.
rich or poor. the current system excludes many people too.
Public lands and access to public lands can't be a playground for the rich and entitled.
The reason they won't do it is because it's more complicated to implement and people will complain in the media about govt "gouging" residents.