As far as whether or not training can "fix" a dog with SEVERE issues is complicated. In short, the answer is "yes and no." And owners that have dogs with severe issues need to understand realistic expectations. We are dealing with two forces here: nature (genetics) and nurture (the way the dog is raised/experiences the world). Nature and Nurture have powerful impacts on a dog and its behavior. Things like aggression, fear, anxiety, etc can be genetic. They can also be learned. Often, a dog's behavior is a result of the combination of the two. However, too many people are quick to say, "Its genetic- my dog is just like that and there is nothing I can do about it!" I would disagree with that in about 98% of cases because proper nurture (raising the dog) can overcome a lot and help shape good behaviors. Other people are too quick to say, "Its all in how you raise the dog!" I would disagree with that 100% of the time, because while proper nurture (raising the dog) can absolutely shape good behavior and overcome some poor genetic influences, ignoring genetics is naive- genetics will always impact behavior to some degree. However, just because a dog has poor genetics and/or has had some terrible past experiences does't mean the dog cannot improve... because in about 98% of cases they can. The question is... to what degree?
The truth of the matter is that not all dogs with SEVERE behavioral issues can be 100% rehabilitated or "fixed." Most dogs with severe issues can make huge improvements in their behavior and become more well-balanced, confident, and trusting dogs overall... to the point where many people will claim, "Fido is completely different now... he is like a NEW dog!" Fido may certainly be more well-balanced, confident, and trusting; however, responsible trainers and owners will acknowledge that Fido's "triggers" (issues) will always be there to some degree, even if it is to a minuscule degree, and therefore they will always take proper precautions when handling Fido to ensure the dog is set up for success in every situation. Some dogs' "issues" are so severe that even with improvement from training & counter-conditioning, they will always be "management case" type of dogs. In other words, these type of dogs may require more handler/owner support and may have more limitations and restrictions on what they can do and what type of situations and environments they can be in. Example: a dog that has severe (or even moderate in this case) fear aggression issues with small kids: regardless of the level of improvement from training & counter-conditioning, will probably never be a dog that is able to be loose at a five year old's birthday party with 20 little kids running around and screaming. Regardless of improvement, risk outweighs reward.
More times then not, when a dog is dealing with SEVERE behavioral issues, a Board and Train (with solid follow-up support) is the absolute BEST option for the dog. A professional trainer has the skill set and the time to work with the dog for several weeks... giving that dog the communication, rules, structure, exercise and leadership it desperately needs on a consistent basis to begin its rehabilitative journey.