I don't think this is a difficult problem to resolve. The National Parks can designate an area for a limited amount of drone flyers, who will buy a permit (helping with NP expenses.) This areas will be a distance from heavy tourist areas. The times this is allowed can also be limited.
Dead Horse Point State Park in Utah has a very similar policy which works quite well. Quoting from their web page:
Between November 1st and February 28th/29th of each year, drone use is permitted by permit only. Permits cost $10 and must be filled out at the visitor center (open daily 9am-5pm) before operating.
Permits are valid for one calendar day from 9am of the date of purchase until 9am of the date after. Additional permits must be acquired for more days.
Permitted operators must follow special regulations to respect the traditional visitor experience of Dead Horse Point State Park and protect sensitive wildlife resources. These regulations are:
The unmanned aircraft:
will not fly higher than 400 feet
must be kept in eyesight at all times
will not fly over people and vehicles (moving or parked) and remain a minimum of 25 feet from individuals and property
will not intentionally disturb or harass wildlife
will not be flown over park buildings or structures
will not be flown in high winds or adverse weather conditions
will not interfere with park rescue operations or programs
will not fly below the canyon rim
Park staff has the authority to enforce regulations and check permits at any time and violations of the rules can result in a revocation of permit privileges.
NOTE: Because of the high use by visitors and concentration of structures, drone operation at the main viewpoint of Dead Horse Point is illegal. Operators must hike at least a short distance and get away from the developed rim to comply with regulations.
Quite an intelligent policy. Tourists are kept away from the drones, and the noise. Drone operators get to take some pictures of the scenery. The park makes some money, because they will have some expense administering this.
I doubt drone flyers are any worse than pet owners, campers leaving litter and starting fires, etc etc. I think there's something rather nice about drone flyers being able to capture the wonder of our national parks from perspectives not usually seen. I could envision a program where the National Parks get to share the best videos to run in the Visitors Center.
As I say, this isn't all that complicated, and everyone would benefit. My $.02