You are correct, although I would say it is really only one assumption: That abiogenesis is a rare, if not unique, event.
This is hardly a wild or arbitrary assumption, however; at least once you reject, as Darwin did, "spontaneous generation," the idea that life arises from non-life as a regular, mundane process of nature. (Some previous scientists, including Creationists of course, had accepted spontaneous generation. Although the number of proponents had been long dwindling, it was universally abandoned as a result of experiments by Louis Pasteur, conducted about the time Darwin's Origin was published.)
Once you've decided that abiogensis is a complex, and probably halting and gradual process, i.e. not "spontaneous," it becomes reasonably obvious that once life does exist, it would adaptively crowd out, and in fact likely and literally eat, the intermediate stages of any further attempt by nature to create life.
Interesting, however, that while you get this right, our article author gets it wrong, and falsely asserts that Darwinian evolutionists not only do, but must accept "spontaneous generation". (The fact that Haeckel also got it wrong, writing at a time before there was any available term contrasting "spontaneous" generation with abiogensis, does not help her.)
Materialist philosophy is neither new nor scientific, but one of the most ancient superstitious beliefs in the world. The ancient version held that matter has always existed and everything that exists consists of matter. According to the modern version, invisible dead-matter spontaneously generated itself from nothing, and then by way of evolution magically produced everything else. To believe this is to believe that the nothingness within the magician’s hat spontaneously generated the bunny.If evolutionism was a gas-powered generator, then spontaneous generation would be its indispensable fuel, admits Ernst Haeckel, pantheist mystic and ardent defender of Darwinism.
My comment was directed to your assertion that darwinian evolution isn't even concerned with the origin of life. If that's the case then why, for example, do Darwinists object when someone postulates a complex genome as a starting point?
Without a specific concept of abiogenesis how do you know that the first reproductive cells were simple and then became more complex, or that multicellular organisms descended from unicellular ones, or how would you distinguish a fossil sequence that is the result of multiple abiogenesis events separated in time from one that is the result of ancestral lineage?
Since you answer to these questions necessarily follows from your particular assumptions about abiogenesis, there is a logical connection. Separating evolution from the origin of life is not logically coherent.
Cordially,