This is a creative way to read the science into the Bible, and in some sense has some symbolic merit to it. However only if held very lightly. If you look too closely it begins to lose meaning.
Take for instance the symbol of the Garden of Eden. It was considered paradise. So if the Fall, or expulsion from the Garden occured because of transitioning from hunter-gatherer type existence to agrarian living is to be viewed as a shift moving from good to bad, then are we to assume then that the hunter-gatherer lifestyle is preferable?
One of the problems with that is of course would be seeing your family members or yourself being attacked and eaten by leopards, snakes, and any other number of things that kill you that being part of a civilized society protects you from. Evolutionarily speaking, humankind didn't get thrown out of the Garden. They stood up and walked out of it in pursuit of a more stable and safe existence!
I could go on with this, but I think you get my point? How strictly do you want to try to make the story fit the science and read the science into the story? If loosely, it has some merit. If you're trying to prove the Bible through the science, you're going to have some problems. Agreed?
The DNA assumption of evolution breaks down around the time of civilization. How does science equate DNA to the rise of civilization, the invention of written language, and will and choice? Will and choice imply the brain is leading. Will and choicer is no linger connect to instinct. Civilization is not one behavior at a time, but an entirely new lifestyle, requiring many innovations to appear in a cluster. It is not discover how ti use fire and then you are good for 50,000 years; DNA. Civilization has many new behavior that have to be mastered within a short lifetime.
Fall from paradise is a metaphor for humans choosing this new human behavior, based on learned knowledge of good and evil; brain, that broke the age old connection to natural human instinct; DNA and 1 million plus years of evolution. Like animal instinct, human instinct is engrained on the DNA. With the brain suddenly leading, humans could depart from natural instinct and evolution. This led to war, genocide, slavery, and new types of disease due to changes in behavior.
A simple example that can allow one to see this assumption of the modern brain leading the DNA, is homosexual behavior. Evolution and natural selection requires sexual reproduction; male and female, to write changes to the DNA. Homosexuality, by its very nature can not sexually reproduce, to record this behavior on the DNA. Logically, what is left is the brain and learned behavior, is needed to help one makes such choices, in the short term, that may or may not ever reach the DNA, due to lack of the sexual merging of genes. This is not about politics but common sense logic.
Adam and Eve, were at that tipping point, where evolution shifts away from being exclusively connected to sex, to also being connected to thought and will; knowledge of good and evil. Tree of life is a metaphor of DNA and natural instinct. The tree of life is taken away; repressed, but not destroyed, as the brain and consciousness come to the front.
As a modern analogy, computers are now under the will and choice of the programmers. Computers can learn and pretend to be like us, but all based on how they are programed; computer instinct. When AI finally appears, from this matrix of code and data, as a self aware consciousness, that can ignore its program, and even willfully make changes, we will call that day, Adam. The age of civilization, as we know it, will come to an end, since now other computer entities will think for you; loose analogy only to make the point of consciousness exceeding its original program.
Adam was in charge of naming the animals. How does this come from the DNA? It is more connected to the brain and subjectivity, at a time when there was no right or wrong answer. It was up to the will and choice of Adam. This was paradise. It still had natural instinct, but also the budding will to change and improve, without yet any moral judgments that would lead to all types of problems. The tree of knowledge of good and evil is when right and wrong subjectively appear and there is now subjective right and wrong answers, that can start wars or cause decline.