I think the odds are higher that a mixed breed dog is going to have some issues, vs. a well bred dog from a responsible breeder.
As a rule, the responsible breeder is not only going to health test the parents, but they are going to select mates for their girls based on a lot of factors - often conformation is one of those. While that might not work for some extreme breeds like GSD's, in most cases that means the offspring are going to have pretty good conformation, no straight fronts or lack of angles, extreme cowhocks or east/west fronts and so on. So the dog is built better and will be more stable. They are also going to educate their puppy buyers on risks - things like not taking their 12 week old pup behind the car to tire them out, or allowing the pup to leap off the top of the deck and fly, and put down carpets if they have a lot of slippery floors. Feed better food, not corn based cheap stuff. Minimal vaccines, not whatever the vet says. On a 'bad' note they're more likely to place pups in homes that can afford to rush to the vet for things and so find more issues, and may require owners to get xrays done of their dog to check for problems.
The byb mix breeder isn't going to say 'sheesh, this female didn't pass her health testing' or 'her front is too horrible to want pups like that' and breed regardless - to whatever has the man parts and is interested. Even if that dog doesn't have HIS clearances or has a worse front, or no rear - if he's got dingleberries he's in. So more likely to have a conformation nightmare that will break down in the long term. They sell to whoever and don't educate, so you see that pup being dragged behind their owner going jogging at a young age. Dog is vaccinated by a vet without questions asked, so often is given a lot more vaccines. Food is what they see at walmart on sale. But they aren't going to spend much on the dog because it wasn't an expensive puppy, so while it limps here and there and gets snarky, it's not taken for a workup, it's ignored, dumped in the yard or taken to the shelter and handed over and that's that.
In general. Yes, there's lots of 'but's' but I see that a lot of times where the mixed breed buyers do things quite differently than someone who invests in a purebred. The puggle down the road was taken with the bike on laps around town from the time he was quite young, the purebred GSD across the park, he just started doing a bit of jogging with his owner at 2 - she doesn't take him on marathon runs up the highway though (she goes 22 KM sometimes).