When science investigates the supernatural and finds an explanation (one that is compatible with the evidence and is replicable by independent investigators), it ceases to be supernatural and moves into the realm of the natural.

If you hand-wave away skeptical inquiry of a topic, then yes, it can remain within the realm of the supernatural, but that is hardly an endorsement, it simply means you aren't willing to subject the topic to investigation with all the tools at our disposal, or at least are unwilling to eschew tools that have proved unreliable (such as circular logic).

Your argument basically boils down to "The spontaneous emergence of life in a universe consistent with the materialistic and naturalistic worldview is so unlikely that we must attribute it to a cause that cannot be accounted for from within that worldview."

To which I say: "It almost doesn't matter how unlikely the emergence of life is, since if life hadn't emerged we wouldn't be having this argument". This is of course the anthropic principle. Which isn't to say that determining the likelihood isn't important and interesting! Just that it isn't particularly relevant to the question of whether the emergence of life can be explained from within a naturalistic and materialist worldview.

You would have to prove that the emergence of life is not only unlikely, but actually impossible, in order to demonstrate that a non-naturalistic origin is required.

Well, actually, to be thorough, that isn't the only possible avenue of proving your point if view, but other avenues would require more data than we currently have, and would require that data to point in directions that demonstrate the universe having agency and intent of some sort.

You might enjoy the science fiction novel Calculating God, by Robert J. Sawyer, which riffs a bit on this theme:

Somewhat spoilery summary is on Wikipedia:
https://en.m.wikipedia.org/wiki/Calculating_God

To give a hypothetical example of the kind of future evidence that could support your argument that nevertheless doesn't spoil the plot of that novel, imagine that buried within the non-coding and non-regulatory genomes of many different life forms on earth there were a message (Bible Code style). The message isn't identical everywhere, mutations do degrade it, but just as often over evolutionary times scales mutations correct the errors. There is no explicable mechanism for these corrections, that demonstrate ongoing interference for no reason than the preservation of the message. Furthermore, the message gets longer in larger genomes, and has expanded in similar ways in species otherwise not particularly related, and to an extent that cannot be attributed to known mechanisms of horizontal gene transfer.

Further, imagine that we discover life elsewhere in the solar system (for example on Titan, or Europa). These extraterrestrial life forms don't particularly resemble Earth's, differing in to he chiralitt of their protiens an do sugars, using sightly different genetic coding and transcription mechanisms, but buried within their own junk genes there are, astonishingly similar messages encoded differently, preserved and expanded upon in sync with Earth's over eons, without there being any explicable means of sharing that data and no other between classes that are otherwise incompatible in encoding, metabolism, or basic chemistry (Titan's life uses methane and ethane as solvents instead of water).

That would be a pretty convincing argument for their being a "supernatural" entity being responsible for creating life and chivvying it along. Whether that entity is also responsible for the creation of the universe is a separate argument, but perhaps there is further evidence somehow encoded within the fabric of spacetime's various universal constants (eg. Contact buried a message deep within the digits of Pi).