But that the first cell just randomly happened in the primordial soup - that looks extremely unlikely,
I don't understand this. Aren't cells just spherical structures that would form naturally from hydrophobic molecules suspended in water/tide pools? That seems likely to me, but I have no background in chemistry/biology.
Nick Lane argues in "The Vital Question" [0] that simple cell membranes are not enough for complex life and complex membranes may have evolved in matrixes around hydrothermal vents.