Argument set forth in this question
The general form of this argument, "X is subject to persistent disagreement, therefore not X" is a non-sequitur and obviously false. (If we accept it, then as you point out, we can use it to prove both "not X" and "not not X", so proof-by-contradiction it cannot be so.)
However, there's perhaps part of the argument structure that you're missing, which is something like:
- Y := If X, then almost all people would find X self-evident with sufficient reflection, study, and examination.
- There is persistent, substantial disagreement about X
- Therefore, not X, or not Y
Claim 1 is usually important to the moral realist, because one would like moral facts to be discoverable by some means, through moral philosophy or inherent knowledge or etc. So the moral realist has to either disagree with the argument structure (perhaps other moral philosophers simply aren't examining correctly), or accept that moral facts exist, but that they are, in practice, inaccessible even to other moral philosophers, which borders on anti-realism.
For the moral anti-realist, not Y is "If there are no moral truths, people might still believe that moral truths exist". Which is not a difficult thing for a moral anti-realist to agree with--it exactly describes the situation they believe exists right now.
Sampson Paper
However, the argument you set out isn't exactly the same as the argument Sampson is using. To quote pg 19 from the paper as an example, Sampson paraphrases Leiter as setting out this argument:
- If there is long-standing, intractable disagreement among the (supposed) experts
about a subject matter S, then the best explanation for the existence and character
of that disagreement is that there are no facts about S.
- There is long-standing, intractable disagreement among the (supposed) experts
about ethics.
- Therefore, the best explanation for the long-standing, intractable disagreement
among moral philosophers is that there are no facts about ethics.
Note that (Sampson's paraphrase of) Leiter's 1. is ~essentially the contrapositive of the earlier 1. So if we apply the straightforward approach, "S := moral anti-realism", we get:
- If there is long-standing, intractable disagreement about whether there are no moral facts, then the best explanation is that there are no facts about whether moral facts exist.
- There is long-standing disagreement about whether moral facts exist.
- Therefore, the best explanation is that no facts about the existence of moral facts exist.
To the anti-realist, this may be totally acceptable, since it forces the moral realist to argue that moral facts may or may not exist, but we have absolutely no way of knowing of their existence, or what they may be if they do exist. Which is not a particularly satisfying conclusion to the moral realist!
I haven't watched the video, but Sampson's argument in particular is actually in levelling the argument against epistemology:
- Supposing we accept Leiter's argument in general, if there is long-standing, intractable disagreement about metaphysics and epistemology, then the best explanation is that there are no metaphysical or epistemological facts.
- Suppose there is long-standing disagreement about metaphysics and epistemology.
- Therefore, the best explanation is that no metaphysical or epistemological facts.
- Claim 1, "if there is long-standing, intractable disagreement..., then the best explanation is..." is an epistemological claim, of which there is no fact.
- Thus, if we accept Leiter's argument, we must be accepting it without knowing (or being able to know) that its premises are true.
- We should reject Leiter's argument.