It appears that from a naturalistic perspective, the only adequate explanation for the emergence of life is that there was a "supernatural" cause.
Nope, you are wrong. A number of ideas and partial theories have been proposed to explain the origin of life, and the investigation of some of them has lead to robust and substantial scientific results (in the normal serendipidous way of science) but they have not so far resulted in a complete, successful or consensus theory. In short we do not know how life originated, so (apart from researchers, experts, or interested amatuers arguing for or against one or another of the "proto-theories") few evolutionists will "insist" that it happened in any particular way.
Furthermore I am pretty certain that most or all evolutionists here would specifically reject "spontaneous generation" as among the least likely explanations, as this term is generally taken to refer to a relatively "sudden" process, or one that is a a mundane (ordinary, everyday) natural process. The general presumption is that life originated as the result of some more gradual, step-wise process of chemical evolution which would not be considered "spontaneous"; or that, even if the immediate origin of life was in some sense spontaneous, there was a precusor chemical system of some substantial complexity which itself had evolved. (I haven't looked into this in years, but last I knew some very interesting self replicating chemical systems had been created, and I think there was expectation that some of them might have industrial application.)
but of course they have no explanation of how the mechanism of life "evolved" in the first place.
Right. Like I just said. Therefore we don't "insist" as to how life originated. We leave that to creationists.