It boils down to whether you think the name is a pejorative or not, much as with any other identifying word. If you associate the word Redskin with negative things, then it's a pejorative, if not, it's not. If you associated the word Irish with negative things, it'd be a pejorative. There's a lot of phony outrage that is nothing more than transparent status signaling.
I think the issue is more complicated and less phony than you think it is. A great deal of what leads to prejudice takes place in the subconscious and is based on the formation and confirmation of stereotypes.
What I regard as phony is much of the outrage about this and other issues, as I believe that a vanishingly small percentage of people actually associate the name Redskins in any sort of pejorative way. As to stereotypes (be they positive or negative, about people, dogs, horses, whatever thing you happen to be considering) they don't really remain stereotypes if they're obviously false. Sooner or later, reality, statistics, experience, observation, etc, will trump false beliefs. Most people can only maintain false beliefs in the absence of evidence. It's difficult to maintain that something is true when all (or the vast preponderance) of evidence is saying it is false, although where ideology or creed is involved, people can and do believe things that can't withstand scrutiny all the time (or at least they say they believe them, often times for the sake of in-group status,) but back to the 'Skins, the name Redskins has nothing to do with denigration of a people, or of negatively stereotyping them. These days, Native Americans are portrayed positively, often as brave and spiritual.
This is not a legitimate case of a group being denigrated, or in any way being discriminated against, nor do most Native Americans see themselves as being denigrated by the name. Native Americans aren't even the ones driving this as a political and social issue. I call it phony because this is an issue that is driven by some white people, who are signaling to other white people about just how good and right-thinking they are according to the fashions of the day.
When it involves a historically discriminated against and segregated racial group I think it creates a complex issue and shouldn't be brushed aside.
That's fine, except there's no actual discrimination or denigration involved here.