Your presupposition, however, is flawed - that the requirements of life are something that had to line up in a row for us to exist. While it's true that we never would have existed without them, individually they are not that unbelievable and governed by simple laws...All not necessarily very special events.
Two points on this:
First, Teleological design arguments usually include the laws themselves when assessing the probability of life existing anywhere in the universe. It asks the question "why are the laws themselves the way they are?". For example, why is the gravitational constant the value it is? If the universe did come from nothing and by nothing, then what governs that these laws are the way they are? This means it is not good enough to simply claim that simple laws reduce the improbability of the events occurring because you first have to address the improbability of the laws themselves.
Secondly let's assume for the sake of argument that all the probabilities when taken on their own aren't that unbelievable. Even when doing this, it still doesn't show that the culmination of all those events are "not that unbelievable". For example consider the rolling of a die. Taken in its own, the probability of rolling a six is not that unbelievable but rolling a six, six times in a row is a 1 in 46656 chance. This means that even if you are able to show that events when taken on their own are not that improbable, you still have to show that the combination of all of them is not that improbable.
Personally I don't get the obsession with the supposed "slight chance" of us existing, because it clearly wasn't slight enough - we're here, aren't we?
Again, this only works if you are presupposing a sort of naturalistic explanation. Flipping back to the poker parody, I could state:
"Personally, I don't get the obsession with the supposed "slight chance" of my getting 100 "royal flushes" in a row, because it clearly wasn't slight enough - I got it didn't I?". One could respond "yes, of course you got it, but I want to know how you got it based on the ridiculously high improbability of it happening by chance alone". The point is that chance alone does not seem to be a satisfactory answer. This is why mutiverse theories which have an eternal past have been put forward.
What? You're very good at obfuscation. And no, the arguments generally go "we don't know how this happened, the only possibility we see is through God, therefore God."
If the above is how you see the god-of-gaps argument, I'm not sure why you have such a problem with it. Essentially from what you have posted I see someone saying "We're not sure how to explain how this happened with our current models. God is currently the best explanation that fits the data and based on this we're proposing that God exists". How is this sort of thing different from any hypothesis (for example proposing the existence of elementary particles such as the higgs boson)? I'm genuinely interested in how you separate these two.
If I was to guess (and so point out if I am incorrect) I would say that you would claim that the only valid hypothesis are those which can be scientifically demonstrated or proven. If that is the case, what happens when we reach the limits of scientific testing? I'm thinking of early stages of the universe or other scenarios where our knowledge bottoms out. What are we supposed to believe today in the cases where science is unable to provide a satisfactory answer and a supernatural explanation fits the data? There seems to be a point at which you're discounting supernatural explanations just on principle.
Because the perspective of science (if a process can have a perspective) is "we don't know how this happened, but let's try and find out, not ignoring all apparently impossible ideas".
But clearly you are ignoring some apparently impossible ideas -namely that of the supernatural. You seem to be ignoring these because you assume that natural explanations are the only valid ones. Is that a fair description? If so, I would challenge you to defend this position.
How is this perspective unfalsifiable? Certainly less so than the presupposition of God, which can never and will never have any proof either way.
Well it depends on what you regard as proof doesn't it?
Certainly I don't think we'll have any proof in the mathematical sense. However, I think we can have sound philosophical arguments (whose premises can rely on scientific data) which could give us probability to work with.
What I meant is that if you always assume that the science-of-gaps (the belief that there is always a scientific explanation available that will eventually be found) is the preferable route to take, then it is unfalsifiable in the sense that it assumes that there is a scientific answer even if there isn't one - it will continue to be pushed forward as truth even if there is never a satisfactory scientific answer found.