At what timestamp does he mention Brandom? I don't remember Brandom coming up in the video (though given the context I did almost hear 'Brandom' when he talked about 'rebranding' the redundancy theory of truth as the deflationary theory). Brandom's anaphoric/prosentential theory of truth is definitely a version of deflationism but I don't know if Blackburn specifically had Brandom in mind when he mentioned deflationism at the end (again, unless I missed that moment or there's a longer version of this video?).

In any case, I don't think your worry applies to Brandom's account of truth. Yes, he does take the content of our thoughts and utterances to depend on how those discursive acts make explicit norms within larger social practices. However, those are specifically practices of giving and asking for reasons (his "deontic scorekeeping") and the structure of that scorekeeping in the cases of science, morality, and a host of everyday topics - e.g. what food is in the fridge, to take Blackburn's example - is specifically one of representing objects (his "de re ascriptions of propositional attitudes"). Those two features alone easily makes room for a minority of moral activists in a community to be right and even for an entire community to be wrong, since by making claims that answer not just to one another (in a community) but to objects they specifically point to limitations in individual perspectives on the truth and even to all of the perspectives the community has so far.

None of this would look like "tapping into something" in a sense that looks like a direct intuition of the moral good but it would involve responding to the objective moral features of the world, just in a way that involves a fundamentally perspectival access to those objective truths and a need to arrive at that truth by learning from the perspectives of others (including perspectives that no one in one's community has yet reached).