Personally, I don't understand the attraction of dog-aggressive dogs. However, I'm not saying that people can't enjoy them - I just don't want one. I don't have the desire (or endurance) to manage dogs who might at any given time decide to rip into eachother. Just not something I enjoy in any way.
I don't honestly know if it is possible (or desirable) to breed out DA in breeds who tend towards it. If I were to try to, I would select proactively towards dogs who have all the traits I want, and minimal DA. It can't be taken out in a single generation, possibly not in 20 generations. But if it were one of the essential traits you wanted to watch for in your breeding program I am certain DA could be reduced without damaging the breed. You can NEVER breed for one trait alone, forgetting other traits. That will ruin any breed.
I'm going to make a comparison to Wolfhounds (who are not DA at all). The issue in Wolfhounds is longevity. Like DA it is so inherent to the breed that you can't just wipe it out by culling a line or two.
There are some breeders who have focused on longevity in their breeding programs, with some success. However, with such focus on longevity the overall quality of the dogs tends to be lost. It cannot become the defining characteristic in the choices as to what dogs to keep or cull. Longevity is a plus - something you want IN ADDITION TO the breed characteristics you are aiming for. When comparing two equally nice dogs if one comes from a line where the dogs tend to get cancer by 6, and the other has parents/relatives who have lived past 10... well choose the dog with the healthier, longer-lived lines. But if the dog from the long lived-lines is inferior to the dog with lines that have a higher incidence of cancer... well don't make the mistake of losing your quality/type.
Of course, you want the best of both worlds. Sound, healthy dogs who live a long time, and who have the breed traits/type you want. You also want enough variation in the bloodlines to avoid recessive traits surfacing and causing problems.
Same goes with breeding out DA. If you have three bitches from equivalent lines, with equivalent types and one is severely DA, one is DA in specific situations, and one is not DA but doesn't have the performance traits you want, then choose the dog that best meets your needs. The moderately DA dog might have the traits you want, and possibly throw less DA in her puppies. When selecting a breeding "keeper" from the litter, try picking a dog that shows the personality you want, but not necessarily the one with the most aggressive personality.
I think it would be possible to reduce DA with a very specific and carefully planned breeding program. But I don't think many people honestly want to do it, nor do I think that if can be successful without open and honest interactions between all the breeders involved. The breeding world is so convoluted I can't see this easily happening...
Anyway, I personally choose dogs who are not DA. While I have had some experience with DA dogs (mostly in rescue foster situations) I honestly have little to no tolerance with it.
If my own mature, non-DA dog suddenly became DA, I'd need to try to figure out why. Yes, I'd manage it as best as I could, but if it came down to it I might choose to euthanize or rehome for the safety of my family. I can't risk my kids being caught in the crossfire of two fighting dogs. Not worth it to me, not even slightly.
But that's why I prefer non DA dogs, and would never deliberately choose a dog who was likely to have that kind of issue.