Are there any behavioral, health, or other benefits to getting a mutt over a burebred dog?
Yes and no.
With a "mutt", you get what's referred to as hybrid vigor, or outbreeding enhancement.
With a purebred, you can see a concentration of "desirable" breed traits that cause problems for the dog itself.
For example, push-faced dogs are more prone to breathing issues (and sometimes eye problems, if they're a bulgy-eyed breed like pugs). Keep breeding push-faced dog with push-faced dog and eventually you'll end up with a litter of puppies whose breathing is so obstructed that they suffocate.
This is 9th grade level science class. If you have parents with poor vision, you'll tend to have children with poor vision. If you have parent dogs with skulls too small to fit their brain, you'll have puppies with skulls too small to fit their brains.
In general, most purebreeds with be less healthy, with more medical complications and shorter lifespans than mixed-breed dogs.
The benefits: you know exactly what you're getting.
If you want a snuggly companion for your apartment, a Cavalier King Charles Spaniel is perfect. If you have a huge yard and go for 10km runs every day, then a GSD or a sibe are more your game. With a mutt, there's a degree of trial and error before you figure out exactly what kind of dog you have.
Ultimately, every dog is capable of being a perfect companion for someone. It's just important to recognize what it looks like when that perfect person isn't you so you can help that dog on their way to finding their perfect forever home.