This is a fascinating question, but one that is beyond my competency to really answer. I am impressed by the fossil record that shows how single-celled life began by 3.6 - 3.8 billion years ago, and considering that the Earth formed 4.6 billion years ago, this hardly gave conditions enough time to cool off so that liquid water could remain stable on the Earth's surface. The Moon may also have been formed by an impact between the Earth and a large body during the first billion years, resulting in the surface of the Earth becoming molten, so this gives even less time for organo-synthesis to have started. There were still bombardments by large asteroids occurring, and some years ago I recall reading how one paleobiologist speculated that life actually tried to start perhaps 3 to 6 times before it got a firm toehold. Each time, it was wiped out by a devastating impact far larger than the one which may have occurred 65 million years ago, wiping out the large dinosaurs.
I personally believe that the synthesis of life on the Earth is an indigenous and inevitable process given the chemical soup that no doubt existed on the Earth then. Chemists have a ridiculously easy time of artificially creating pre-biotic organic molecules from the primordial atmosphere and water of the ancient Earth. There have been speculations that most of the organo-synthesis occurred inside comets that also bombarded the Earth 4 billion years ago, but I have a major quibble with this scenario. Chemical reactions occur at a rate that depends on temperature. Any cometary reactions at temperatures near 20 - 100 K, would take MUCH MUCH longer than identical reactions on the surface of a nice warm planet that just formed. Although astronomers can look into distant interstellar clouds and see astonishingly large molecules, these are still quite small compared to the amino acids needed for pre-biotic RNA and protein synthesis. The comets may have transported a rich molecular soup frozen onto dust grain surfaces, but such a soup could easily have been synthesized on the Earth in a lot less than the millions of years taken by the chemistry of interstellar clouds and cometary bodies. Scenarios that attempt to synthesize whole DNA molecules inside comets have an even tougher time doing so because of the low temperatures involved on the surfaces of cometary dust grains.