First, life came about rather shortly after the planet was able to sustain it. So this timing is either very unlikely, or biogenesis has a high probability of happening, or the initial life got seeded from off planet (from a very rare initial phenomenon).
Second, all the biosignatures from all forms of life are similar. For example, corality (left handed versus right handed molecules) is the same in all life examples we had. So that means that biogenesis happened only once. Otherwise we would have multiple unrelated examples (yes, there is the possibility that one form "ate everything else", but we have all kinds of variety instead of only one organism becoming dominate). So if biogenesis happened only once (for all of our bio examples), then it is either a rare event that just happened to occur very shortly after the planet cooled, or it happened sometime before Earth was hospitable for life and elsewhere in the universe.
Does any of this make sense? Or am I reading too much into the tea leaves?