What is "the reliability of the claim itself", if it is not the reliability of the method that is used to estalish that claim?
> Given that your argument makes no reference to what is more or less probable
Yes, it does. Reliability (of a claim) is a scale expressing a probability of being true. "reliable" is the high probability end of that scale, "unreliable" is the low probability end of that scale.
> (but then you'd need to establish how those probabilities were derived)
Yes, I have, over, and over, and over, even in my last post.
> You can't/don't rely on scientific claims "_for_" anything?
That wasn't the question. Obviously, the more likely a claim is true, the more you are justified to rely on it as the basis for making decisions about the future. But the phrase "claim A is reliable" makes perfect sense without a "for X", simply as an expression of how likely the claim is to be true.
> As in, the claims of science are not reliable for things like putting satellites into orbit or pursuing further truths, they are just likely a correct description of reality?
Being a correct description of reality is what makes a claim useful for making predictions about the future behavior of that reality, and thus for influencing reality in such a way that a satellite does behave as intended/predicted.
> You're shifting the burden of proof.
No, I am not, I am simply performing the empirical observation that is required to establish whether the claim is testable.
> Not having something to accept as falsification for the claim "water only boils at 100°C or greater (under 1 atm)" doesn't mean that there can't be one.
Actually, that is all that testability is about: Whether there is something that would convince you that you are wrong if it were demonstrated to you. For "water only boils at 100°C or greater (under 1 atm)" that should be the case, because that statement under standard English language rules has semantics that imply a contradiction with certain potential observations. But ultimately, what counts is whether whoever is making the claim would accept those demonstrations as contradicting their claim.
> (it being non-falsifiable something we seem to agree on since we think it is self-evident).
No, not at all.
The problem is that that whole question is massively ambiguous, and the answer is completely different depending on the interpretation, or possibly some interpretations are just nonsensical questions.
If you mean whether our senses correctly reflect "ultimate reality": That would indeed be unfalsifiable, and for that exact reason it is indistinguishable from shit someone made up, and thus a highly unreliable claim.
If you mean whether our senses generally produce a coherent picture of an outside world: That is falsifiable, in that we can cross-correlate different senses, between perceptions of the same sense over time, as well as between our own and others' perceptions. I am guessing that that is maybe why you say that this is self-evident, but really, it's way more than just self-evident. Actually, that is exactly how we do know in how many ways our senses indeed are not reliable. There is a lot of redundancy in our senses (and indirect methods of perceiving the world) that allows us to test the reliability of each sense, with no sense in particular being one that we just have to trust blindly.
But mind you that all those tests of our senses do not in any way demonstrate that we can sense "ultimate reality". Possibly, we can't, but that is not a problem for our ability to navigate the (perceptual) reality we find outselves in, or for the testability of that fact.