Hi Gregg. The Muir trail is what we locals called "The Freeway". I avoided it as much as possible, although have incidentally covered most of it, section by section, as part of unavoidably getting certain other places off-trail themselves. My younger backpacking understudy of the past decade did the Muir Trail in autumn to avoid people, taking six weeks, enduring three blizzards, and hauling a 110 lb pack in order to alleviate resupply issues, which are unrealistic that late in the season anyway. Lots of Muir / PCT thru-hikers are in such a hurry trying to be on schedule at resupply points that they get downright Zombie-fied, and don't even notice all the beauty around them. They're just clocking time. Therefore, given the unusual circumstances last summer, with both understaffing issues due to an epidemic, and repetitive severe forest fires, things got pretty clumsy with their booking system, which went remote, online, to half-baked emergency Federal site. But that has improved.
Ordinarily, for any given trailhead there are entry quotas for sake of keeping the backcountry still pristine. For any given day, 60% of the wilderness permits are available by reservation, and 40% available on a first-come walk-in basis no earlier than the day before, at the particular FS or NP office responsible for that jurisdiction. But with no walk-in's allowed either last summer or apparently this one, there's a kinda clumsy online substitute. Highly popular trails like in Yosemite NP are of course going to have the most demand, and therefore reservations fill up long in advance, while many other places in the high country have relatively little pressure on them, and are easy to get permits for, or even lie outside controlled areas. At the end of Sept (or end Oct in the case of Yosemite NP) permits are no longer required until the next subject season. But by then, most roads up to the handful of high road passes are closed anyway.
Keep in mind that, just because Yosemite Valley itself is a very well known tourist destination, just like Yellowstone or the Grans Canyon, significant parts of its high country as nearly vacant and largely devoid of trails. Same goes for Sequoia NP, and especially Kings Canyon, as well as many of the numerous designated Wilderness Areas. A lot of that topography is extreme enough to be self-protecting, at least unless something like a massive dam project is contemplated. There are many many many beautiful uncrowded locations, especially for the determined hiker, but also even for the car camper if they do a little homework and avoid the most obvious options.
Wilderness permits are only required for overnight or prolonged backpacking use. Dayhikers are exempt. This system lets them monitor how many people use particular places over the season, but also is important in case rescue is needed. All hell broke loose last summer when the rescue system was utterly overwhelmed by massive fire and smoke issues. Assistance from military chopper crews was required, and it took up to a week to get most people out. I know some of those people - smoke inhalation issues, catching covid either on the choppers or in evacuation facilities afterwards, really a mess.
The other nice thing about the Sierras is that dedicated OHV and 4WD are provided for those enjoying that kind of activity, in order to keep them away from sensitive areas. These are classified according to difficulty, including potentially fatal routes if they aren't scouted on foot in advance, with a 2 mile per DAY speed limit, and requiring winches and convoys. In other words, you could walk them faster, but some four-wheelers want a real challenge.