The Garden of Eden Did Not Have a Latitude and Longitude
I don't know enough about geology to evaluate the dry-Persian-Gulf theory, but I think on purely textual evidence the theory falls down. Genesis 2:10 states "A river watering the garden flowed from Eden; from there it was separated into four headwaters." This pretty clearly places the garden near the source of the four rivers, not near their downstream confluence. The mountains of eastern Turkey thus seem a more likely setting for the garden of Eden, since that's where the headwaters of the Tigris and Euphrates are.
As an atheist, I don't believe there was a garden of Eden -- or even necessarily any specific real-world place whose legend got embellished into our garden of Eden story. But we can certainly ask where the authors of Genesis thought they were placing the garden. Trinklein's search for the garden of Eden goes astray, however, by imposing a very modern conception of "where" onto the Bible's authors.
Trinklein says in the linked video that the fact that real rivers are used to give the garden's location is good evidence that it was real, since that would allow people to go and check to see if the story is true. And therefore, we can use the coordinates given by the Bible to locate the garden for ourselves. This line of thinking makes two mistaken assumptions.
The first mistaken assumption is that the clash between belief and skepticism would have made sense to the authors of Genesis. Trinklein certainly approaches questions of Biblical interpretation from the mindset that there are nonbelievers out there, against whom the Bible's claims must be defended. But that would not have been the approach of the authors of Genesis. In their world, belief in one myth or another was taken as given -- evidence-based skepticism is a modern invention. Indeed, Genesis itself does not present clashes between God's people and followers of other gods as clashes between two alternate theories of the divine, only one of which can be accurate. Rather, the other gods do actually exist -- they're just not as powerful or worship-worthy as the God of the Bible. There was no fear of a 5th century BCE Richard Dawkins heading out to explore the headwaters of the Tigris and disprove the existence of the garden of Eden. Moreover, even if a skeptic had wished to seek out the garden of Eden, it would have been an arduous journey. Few people in the ancient world would have the means to carry out such an expedition.
The second mistaken assumption is that the authors of Genesis and their ancient audiences would have thought about geography in the same way as we do. I recall voraciously reading Genesis as a kid, carefully comparing the events in the story to several maps that were included in my Bible, as is standard for modern editions. (Indeed, I recall being particularly annoyed that I could not locate the Pishon and Gihon on these or any other maps!) Packaging these maps along with the Bible implies that we should view Bible geography the same way we view modern geography -- as a matter of events taking place in defined locations on a roughly-spherical Earth, which can be viewed from an "objective" bird's-eye view. But this way of looking at the world, and the proliferation of spatially-accurate maps that accompanies it, is really a product of the modern era (albeit with significant ancient Greek antecedents). The few ancient maps that we have show clear signs of being conceptual cosmological diagrams, with little concern for fidelity to latitude and longitude. This was not merely a reflection of poor data sources and mapping techniques. It was a reflection of the way people actually thought about a wider world that they would likely have little reason to ever travel through. The world around them was organized into a symbolic system, inferring the existence of features like an encircling sea, Mountains of the Moon at the source of the Nile, or other entities that nobody had ever actually seen.
Let's apply this idea of geography as symbolic cosmology to the garden of Eden. The idea of a single central source for the world's rivers, paralleling the single ultimate destination, is a common one, giving a simple organization to our understanding of hydrology. The garden is placed at this source to remove it from the mundane nearby world. Moreover, there is a clear symbolic link between the idea of the garden as the origin place of humanity, and the garden's location at the headwaters of four life-giving rivers. These symbolic cosmological ideas would have been far more important to the writers and readers of Genesis than any concern to send out an ancient Lewis and Clark to map out the true location of the Pishon's source. Thus Trinklein's spatially-accurate maps -- which show the four rivers having origins far from each other but flowing together on the bed of the dry Persian Gulf -- misleads us into using spatial logic to seek the location of the garden. We may not be able to find a latitude and longitude for the garden because its coordinates are given not in degrees but in conceptual connections.