Why Yellowstone doesn’t see more backpackers, and enjoy the not necessarily direct increase in hype, eludes me. The usual dark service, I mean park service, bureaucracy surely has something to do with it. You can’t bring your dog, and you’ll need a permit. Mister Grizz is also likely at fault. I live and weekly visit another Grizzly population center, the Glacier/Bob Marshall complex, but I consider bear sightings in Yellowstone both more likely and more potentially problematic. A long snow season and high river crossings shortening the so-called hiking season makes Yellowstone less amenable for backpackers, as do the often biblical mosquitoes. Lastly, the enhanced experience of traveling in the park during high season along with nearly a million other humans a month should scare you. Erratic driving during a bison jam is more likely to cause damage than bears, bugs, or hypothermia.
Nonetheless, backpacking is Yellowstone is a really good idea. Yellowstone enjoys a lower level of backcountry visitation relative to frontcountry numbers when compared to parks like Yosemite, Glacier, and the Smokies, and permits are as of this writing still free. A modest 20 dollar US fee for an advanced reservation is almost guaranteed to get you whatever route and campsites you want. Most backpackers stick to a few, well traveled areas, and thus even in the height of summer, you might go days without seeing anyone.
Speaking of campsites: the designated backcountry sites in Yellowstone are the best in any park I’ve visited. Unlike Glacier, Isle Royale, or Canyonlands, where many of the camps feature dusty, crowded, rock-hard dirt tent pads with poor views (let us not even consider shelters in the Smokies, and the joys of mice on your face), sites in Yellowstone are for one party only. Most feature only a sign and a pole for food hanging, leaving shelter location to your discretion. You even get to dig your own cathole, so it counts as actual backpacking. Many sites are so infrequently used, the spur trail to them is all but invisible. Most are located in better places than a first-time visitor would be able to select on their own, hiding at the edges of stunning meadows or gravel beaches so sublime your imagination will be melting out your ears.
Mister Grizz is indeed something to worry about. Do your research beforehand, and come to your own conclusions. You might decide that walking amongst a full complement of megavertebrates has a charm which cannot be equaled. Listening to bison mating calls in July is certainly an experience which cannot be compared to anything.
Yellowstone is above all else a park for backcountry gourmands. The caldera which makes up all of the park itself may be ringed by impressive peaks on almost all sides, but most of the ~thousand miles of trail within the park traverse gentle river valleys, broad ridges, and rolling hills. It is not scenery which grabs you by the lapels. It is scenery which requires a solid week of miles and many hours meditating upon tea steam and at-last warm feet to understand.
On my most recent trip there I spent a night in the middle of our trip talking with an archeologist around a campfire. He and his team were out studying the 1877 flight of the Nez Perce across the recently-created park. We agreed that it was in many ways an artifact of an earlier era that the park had been created at all.
Yellowstone may have the highest concentration of thermal features on earth, but most of the park, and the vast majority of the current backcountry, is not capital S spectacular, nor capital M national park material. What is it is gentle, fertile terrain which is easy to travel through and thus supports prodigious amounts of wildlife. The geology of Yellowstone is unique. It’s position as a high gentle plateau is as well. What is most unique is that such country still exists in many respects unaltered, and fairly accessible. Enjoy it.