There is just enormous amounts of shoddy thinking out there from people on the subject of Origin of Life. I'm particularly annoyed by the non sequitur "the universe is large, so there must (with high probability) be life elsewhere." (If anyone reading this thinks that's a valid argument, go look in a mirror and slap yourself.)
That argument is basically "there is a value of N such that for any p > 0, N p is much greater than 1."
But that's obviously wrong. For any N, there are values of p > 0 that make the product N p arbitrarily close to 0.
The dim intution behind the argument was that p can't be "too small". But given our current understanding of OoL, that's not a justified assumption. p could be exponentially small, if OoL requires some extremely unlikely step.
Natural selection is great once the system's reproductive fidelity is good enough to support it. The problem is bridging the gap from small molecules to that system. The smallest system we know of that can independently support Darwinian evolution has billions of atoms.
What we want is the probability for at least one other place other than ours to have life. This would be 1 - (1-p)^N, which does tend to 1 as N gets arbitrarily large.
To get that formula: (1-p) is the probability that life does not exist in a place, so (1-p)^N is the probability that ALL places where life is possible, has no life. Therefore, 1-(1-p)^N is the probability of the opposite of that (where at least one place has life).
Furthermore, proponents of an extraterrestrial origin of life on Earth will doubtless argue that nearby life may have had a common origin.
However, if we found life on Mars that same Bayesian reasoning would imply a meaningful lower limit on p as well, since life on Mars is independent of our existence to observe it.
If we found life on Mars tomorrow, how well-defined would that lower limit become?
Just finding life on Mars that's the same kind of life as on Earth would not tell us much, as it could be explained by panspermia. There are Mars rocks on Earth, so transfer of life in those rocks should have happened constantly. If early Mars were habitable it almost certainly had life, due to this transfer.
That the probability goes to 1 as N goes to infinity FOR FIXED p is just another example of assuming p can't be "too small". The probability also goes to zero as p goes to zero. Why are you fixing p and not N? Why are you assuming p is large enough that N is in that asymptotic range where the probability has approached 1?
The analogy I like here is those "collect the letters" games you see at fast food outlets and grocery stores. Buy a Happy Meal, get a scratch off ticket. If you collect all the letters in some phrase you win $N million. When you start the game, the trend is great. Letters are arriving and the phrase is filling in. But try as you might, that last letter never shows up. The game ends and you've won nothing. Of course, the way the game was designed was that last letter controls how many winners there could be. All the rest were distractions.
I find it weird to use a deliberately rigged game as an example. If one of the previous letters was wrong, the last letter being right means you don't win either. It's like saying the difficult steps are going to be extra difficult because other steps were found easier than expected.
You could argue that the priors were garbage I suppose. I'm not arguing for any particular probability.
The McDonalds example does not have independent variables as X1..Xn-1 are deliberately increased as Xn is decreased. I'd also argue that origin of life doesn't have independent variables. If chemistry turns out to be more or less powerful in one setting, it should do something for our assessment of other settings, especially when it's similar processes.
Of course, if some people don't understand that this one is not an established fact, and that annoys you, I can't say you are wrong.
I agree that it is arbitrary that the dimension of the exponent of n has to be larger than the negative one of p. That probably stems from the assumption that the universe is endless.
You're focusing on (lack of) evidence for a mechanistic explanation but that's not exhaustive.
Also I think you missed my point which is about Bayesian estimation of p, not of N.
Bayesian reasoning (by using the fact that we exist rather than don't exist, as well as other info about our existence, such as how long it took us to evolve) helps us estimate a probability distribution of p, as well as a central tendency estimate.
See e.g. https://www.liebertpub.com/doi/full/10.1089/ast.2019.2149
"exponentially" is not a measure of size, nor is it a measure of relative size. If you think this anything base on "exponentially small" is a valid argument, go look in a mirror and slap yourself.
"Exponentially small" here means "the probability could be ~ e^-n" where n is a number proportional to the complexity of the minimal evolving system. This would happen if there's some gap that has to be bridged by random chance before we get a system capable of sustaining natural selection.
The point here is that this could easily be vastly smaller than 1/N, where N is (say) the number of atoms in the universe x age of the universe x rate at which atoms might interact to form such systems.
I think you could have easily understood this point if you had made an effort to do so, without me having to spoonfeed it to you here.