As the theory of accretion goes, Earth came together from matter in the nebula surrounding the early Sun. The end of this process was marked by giant impacts. One of these giant impacts consisted of a proto planet we call Theia. According to the theory this happened about 4.6 billion years ago. Theia was about the size of Mars is today and because of this impact we have our Moon. The impact also had the effect of heating the Earth to a molten ball that was too hot for any organic materials to exist. If the primordial soup ever existed then it had to form after that event because organic compounds needed for life cannot exist at those temperatures.
This had another effect on the formation of life. First of all the Moon started in a low orbit. This gave the Earth very high tides making for a lot of stirring. The tides caused by the Moon today are but a dim echo of the tides in the early history of the Earth. When the oceans formed these tides roared inland with a great deal of energy; the energy of that tide was by far more energy than the worse tsunami in recorded history. Continental crust formed within 150 million years after the impact of Theia. This led to a lot of interaction between the small patches of land on primordial Earth and the oceans. Soluble compounds on the small patches of land ended up in the ocean. The oceans also started out with large iron content in their chemistry.
Another factor we must consider is the Earth is stabilized by the Moon. The poles stay within a few degrees of a right angle with the orbital plane. This has lead to long period of stable climate on the Earth. This stability is crucial for the development of life.
Because of the collision with Theia the Earth also possesses a large iron core. This iron core is still liquid today and it shields the Earth with an electromagnetic field preventing a good portion of ionizing radiation reaching the surface. This too was important in the development of life on our planet.
There are still many questions that are unanswered. The origin of the huge amount of water on the planet is still a mystery and there is no viable theory to explain where the water came from. The most reasonable explanation would be that the water came from comet impacts. And it is also thought some of the water was dissolved in the molten rock. At any rate we need water to make the primordial soup.
So all this stuff is stirred and out comes amino acids, which turn into proteins, which turn into self replicating cells. This all happened almost as soon as the planet was cool enough, below 300 degrees Celsius. No one can reproduce all this in the lab. It is believed that life was abundant 3.85 billion years ago and life was well established on the Earth. This figure comes from ancient carbon deposits found in western Greenland.
So the oceans of the Earth were stirred and out of that came amino acids. One criticism of this notion is that amino acids come in both right hand and left hand forms. On Earth only the left hand kind is found in life forms. There is no reason chemically to support this discrimination.
There is a common chemical origin of all life found on Earth. Besides amino acids, proteins and RNA all have a commonality that cannot be deemed a coincidence. This commonality points to a single origin for all life on Earth. Given the size of the primitive oceans we cannot claim that the whole ocean was a primordial soup. Different things would happen in different places and these places would be separated by thousands of miles. The same is true of small pools in diverse places. These diverse pools would produce diverse results. The results are not diverse and therefore all life on Earth came from a single source.
A single source pool for the soup is all pretty unbelievable. Given the turbulence of our planet's early history a single soup basin would have to develop life in less than a year; a little pool of soup would not last very long. Another explanation would be that life came from a comet or meteor impact. What soup that came from is beyond our knowledge today. It would be interesting if we could get samples from the comets and asteroids in our solar system. But again, we're talking a single event in the Earth's history. Comets and asteroids may possess organic matter and even amino acids, but none have been shown to have life or even a form of proto life.
No scientist can tell us exactly how or why all the precise order demanded by living organisms came from chaos. There is no defined chemical mechanism that is capable of putting together the order needed for life. The best experiments only put together organic molecules using molecules that come from questionable origins. The processes that came from the diverse sources of experiments have never been married together demonstrating the holes in our knowledge and again, no life has been created in a lab from unanimated ingredients. It is all a mystery which still needs to be solved.
We do know that life is an involved chemical process that is self replicating. We know that life depends of energy from some outside source the one we are most familiar with is the Sun; it can also use geothermal or chemicals found in minerals. We know that all life on Earth has a chemical commonality. We know that life started at some point early in Earth's history. Most of all we know that the chemistry of life is very complex even in its simplest forms. This all infers that a soup cooked up by any process is a questionable source for life to begin.
Many take their belief or disbelief on the primordial soup and use that to prove the existence or nonexistence of God; they are only fooling themselves. Those that believe in God might say that because there was no primordial soup proves God's existence. It proves nothing. If there is a God he created the universe using processes we may be able to affirm and describe. We do not know the nature of the process for which he may have created life. If you are an atheist, then the primordial soup theory does not prove that God does not exist for the same reasons; after all, the process could be that which God used. In the end we must all understand that science is a search for knowledge.
All true scientists should purge their preconceptions for their search for the truth. Currently there are no understandable truths concerning God; and, the nature of God cannot even be described scientifically; the only way we have been able define and describe God is with our metaphysical beliefs. There are no scientific theories that confirm or challenge the metaphysical beliefs concerning God.
As for me, the theory primordial soup explains almost nothing. I am sure there is an as yet to be described process that explains the beginning of life. After all, we all know that there is life on Earth today and it had to come from some source.