I think that your general claim is correct. We certainly have no way to measure the probability of life’s spontaneous generation, but we know that life does exist.
However…
Suppose: A man walks on water, in clear sight of thousands of other people. It is obvious that this is not a trick; perhaps it is verified by experts in science, etc. Is this evidence of supernatural intervention?
This is a good question. The answer hinges on whether a prediction was made before the fact. If someone says, I am going to walk on water, or so-and-so is going to walk on water, at such-and-such a time, and does so, I’d accept that as evidence of supernatural intervention, as humans do not have such knowledge that such a combination of events would happen. If it’s just an after-the-fact thing, some ugly fat slob was seen walking on water, who had given no indication that such a thing would happen, then again it only reflects our lack of knowledge about the initial conditions.
If you say yes, then it seems like your probability calculus has gone out the window. For how is this different from the above scenario?
Because a prediction was made before the fact. Probability is only a measure of lack of knowledge, but a correct prediction would show extraordinary knowledge.
Something extraordinarily unlikely has happened,
Stop right there, because you have missed the point of the OP. “Unlikely” has no meaning except relative to our knowledge or lack of knowledge. If you want to get right down to it, our universe is not just “extraordinarily unlikely”, its “probability” is zero, because it is one out of an infinity of possible universes. Yet our universe is here. The only way that probability makes sense here is in terms of lack of knowledge.
but since it
has happened, it seems that it must be more possible than we expected – surely science will sufficiently explain it (as if that precluded God’s intervention

).
Who is saying scientific explanations are precluding God’s intervention? Certainly not me. But tell me, why is it so improbable God would have arranged the initial conditions such that life would arise via abiogenesis and evolution, rather than monkeying around with creation to ensure it arose? How are you going to assign probabilities here? And the comment about the zero probability of the universe is true whether God exists or not.
If you say no, then you are claiming that, if God exists, nevertheless no tangible evidence could ever materialize, such that we could know of His existence. In which case, it’s no wonder you’re an agnostic.
Every finite effect is explainable via a finite cause. Inferring the existence of supernatural entities is one thing, and one which I could be open to, as I said above. They need not be the philosophical God, however.
And this distinction needs to be made clear, also. Inference can only work with finite sets, and is only relevant to the actual universe. Now, there are a finite number of molecules in the universe, and thus a finite number of possible arrangements. We have no knowledge about what those arrangements were on the early earth, but we know a very specific configuration would need to be the case for abiogenesis. Our existence is evidence though that such a configuration in fact was the case. Saying that it was “unlikely” is an illegitimate reification of probability, which only reflects lack of knowledge.
But some versions of this argument (e.g. the fine-tuning or anthropic principle) arguments are different. They ask what are the “odds” of a universe existing where life can exist. Because there are an infinity of possible universes, and an infinity of possible universes where life can exist, any inference breaks down. Probabilities are fractions (by definition) and fractions (by definition) only apply to finite entities.