That we pit nature against nurture has within it the implicit assumption that what "nature" gives nurture can't change.
DNA is mechanistic in how it operates. Its power comes from our ignorance of the way it affects us. When we know how it works we're able to get in there and change our destiny. "Nature" and "nurture" aren't opposites, they're tightly integrated. To the extent they seem distinct it's because we are ignorant of how they relate in specific instances.
E.g. we might find that a particular gene is highly correlated with the early onset of kidney disease/failure. "Oh no!" says the person who finds out that they have a (say) 90% probability of kidney failure in before they're 60. What if, however, the entire reason that this gene led to kidney failure was because it decreased a person's desire to drink water? Low water consumption then led to stones leading to kidney disease. Given this information, the impact of the gene on kidney failure in an individual is entirely negated, because they can internally correct and just choose to drink more water.
This same line of reasoning can apply to intelligence: what if our genes code not for some mysterious magical "competence", but instead for a predisposition to find certain things interesting? And what if we're able to make the right things interesting in the right way? This enables us to significantly diminish the impact of those genes, too.
Similarly with dogs -- the impact of genes as an animal grows from being a baby to an adult is not on eventual outcome, it's on development. Some dogs are less trainable because they're less food motivated. What if, during the dog's development, we could slightly restrict access to food, and grow in the dog the desire for it? Now the dog is more trainable.