If conditions were just right for the formation of one living organism, couldn't billions of unrelated living organisms have started up at the same time? It is a large planet, lots of room for lots of organisms.
Then some organisms might have had offspring and some might not, and there is the lawn, with many of the individual grass plants being similar but not identical.
Eventually one family line will dominate in an area. We see this clearly in the fall when leaves change in the forest. All the leaves of one kind of tree in one area of the forest will change at the same time, yet the leaves of the same kind of tree in a different part of the forest will change at a different time. You can see this, it is clusters of autumn color separated by clusters of green. Each cluster of autumn color is of related individuals.
So the descendents of one aboriginal organism will carry on, and descendents of another aboriginal organism will disappear. Of the billions or trillions or zentillions of aboriginal unrelated organisms, only a few billion family lines survive to this day.
Can DNA be traced back to a single aboriginal strand of DNA? Consider that some DNA sequences are favored and will appear in many different, unrelated family lines out of the zentillions of aboriginal unrelated family lines.
Mere chance? No, not chance. Some DNA sequences favor survival.
As to the lawn model: Is it likely that there was first a living organism and that everything living descended from it?
My understanding is that fossils generally cannot provide tissue to map DNA information. If that is the case, and if this research project stays away from making projections based on evolution theory, what they'll end up with is a huge database with specific genetic information (which is also huge) on each and every known type of creature - plus a few that are now extinct.
Therefore, I do not see where this can project will have the database to graph a "tree" which has time vertically and morphology horizontally. It would be based on available genetic information.
But using that information, I can see them sorting, parsing and matching the database to look for genetic matches. If the resulting chart is genetic information vertically and morphology horizontally - I personally think it'll look like a "lawn."
Just my two cents...