Doesn't really matter if TOS said it or not. SNW said it's the Flagship, so it's the Flagship. Considering the reverence towards the NX-01, I'm perfectly fine with The Enterprise always being the Flagship, if there's one currently in service. It retroactively pays respect to the NX-01, and I think that's great.
I get that you believe there is a link between SNW and TOS because CBS says so. With that said, it is still perfectly valid to point out that nowhere was it ever mentioned or even implied that this was the case in TOS. Kirk is just a Starfleet captain no different from any other (and so is Pike, for that matter), and the Enterprise is just one of twelve or so ships of its class. It's not a crime to point out retcons.