The real problem is that all the solutions cause other problems, because of factors driving the traffic.
Humans need nature. At first, all of them lived in it. Then only most of them. Now a majority live in cities. The less nature people have, the unhealthier they get. People need more nature than they're getting. So anything that restricts people's access and lowers traffic will necessarily worsen their health.
A crowded wilderness isn't really a wilderness, though. It no longer serves its purpose for wildlife or humans. Wilderness is about peace and quiet, about solitude. To a slightly lesser extent, the same is true for the outdoors in general. Not only does heavy traffic wreck the space, it wipes out the things that people go there for -- the opportunity to relax and get away from others.
Privacy, solitude, and quiet are also survival needs. Losing them causes stress, and stress kills people. It contributes to numerous problems that cause people to suffer and die. This is expensive. Again, lowering access to nature harms people by denying them access to these survival needs.
It's not equal either. Poor and ethnic neighborhoods have much less green space compared to rich and white ones. Increasing green space raises health and lowers crime, and it's urgently needed. It's a win all around.
The catch is, almost all of the proposed solutions for the national parks basically amount to turning them into refuges for the rich. It is absolutely a dominance fight; whenever resources become scarce, the higher-ranked individuals hog them and leave little or nothing for anyone else.
A reservation system completely removes spontaneity, which restricts it to people who have reliable control over their own schedules. A large portion of workers have unpredictable schedules, overwhelming those in lower-paying jobs. A reservation system would mean that they could never get to a national park, because they couldn't simply take advantage of a rare day or few off work.
Any price on admission will shut out some people, but a price high enough to reduce traffic effectively will shut out most people -- first the poor and then the middle class, as the price goes higher.
Also, if you have to pay to use something, it's not really yours. This fuels a disconnection with nature in general and public land in particular. If people don't feel a sense of connection and investment, then the parks become "someone else's problem" and the public won't protect them or the rest of nature. That's already a problem; we can't afford to make it worse. Connection with nature is absolutely essential both for human health and for the environment.
Solutions that would actually help:
Generously fund national parks and other public land using tax dollars. No user fees onsite; you pay taxes, so you already paid for it.
Expand both local greenspace and national greenspace. Act to preserve what we have instead of clearcutting or paving it. The more parks people have locally, the less intense the craving to gorge on faraway sources. The more space available, the more people can spread out, the less crowding, and the less impact on those spaces.
Separate some refuges for wildlife from areas for human use. Wildlife needs places to call its own, just as humans do. Layering the uses will create a slope from zero human use through wilderness with occasional access to parks with higher traffic.
What are some of your ideas?