Not necessarily. A mixed-breed dog does inherit genes from both of their parent breeds and is, therefore, more diverse. However, there is no guarantee that an individual mixed breed puppy will inherit the right combination of genes that makes them healthier.
A mixed-breed dog can take on any combination of traits from their parent breeds. So, a puppy could inherit more dominant characteristics from one parent breed over the other, they could get a nice balance from both, or any combination of traits from them.
A puppy could “win the genetic lottery”, inherit the best of the best from their parent breeds, and avoid some of the breed-specific health issues associated with those breeds. But, it’s called a “lottery” for a reason - there is no guarantee. A puppy could also inherit the worst genes from their parents and be prone to the breed-specific health issues common to both of their parent breeds.