I think what's a bit interesting from the Second Circuit's standard for stay pending appeal is that it looks to "irreperable" injury for the non-movant instead of just "injury," which I don't think is necessarily inconsistent with Niken. That was a pretty unique context, but it seems to me that the difference potentially matters. The injury-to-the-other-party factor almost always is maleable by the party ginning up the opposition to a say, but the Second Circuit's requirement (assuming Torres is still good law after Niken) that the opposing party must show irreparable injury seems potentially to matter here. The NFL has lots of injuries it could claim here, I suppose, in trying to ward off a stay pending appeal -- commissioner's authority, whatever -- but the only possible irreparable injury it could show, I guess, would be an argument that Brady might retire and thus avoid punishment.
Also, what I like about the Torres formulation is that it tries to quantify what it means to have a likelihood of success on appeal, which can be slightly different from what the likelihood of success prong when seeking an injunction. I think there are some circuits, for example, that have noted that "on the merits" is either wrong or must be interpreted super broadly in the stay pending appeal context. "Success on appeal" is often but certainly not always co-extensive with success "on the merits." For example, you could argue procedural error in the district court that doesn't go to the underlying merits that would present a basis for reversal and thus support a stay pending appeal.