It's not purebred vs rescued mutt. It's the age of the dog and whether or not it was mistreated and developed problems that new owners are unlikely to know how to fix.

Young rescues are usually fine.

As you noted, a good rescue agency will do their homework before placing dogs. Not all rescues are good.

And of course, all dogs should be put through a puppy skills course of some sort, even if they're not puppies. Most pet stores and many boarding facilities run classes. They're as much for the owner as the animal.

I've owned three dogs as an adult, the first two from breeders, the 3rd from a rescue. The rescue has had more behavioral issues than the two I raised as puppies. All the behavioral issues are directly linkable to his previous owner (a puppy mill, where he was kept crated most of the day). It's been a TON of work to get him to a point where he's mostly well-adjusted (and I knew going in this would be true). If he had been my first dog, he likely would have been my last as well.