So, do you think when Scotty mentions his "prize beagle" the assumption is that Archer's no longer alive but his beagle is? Maybe if Archer had died a few years earlier or something. Hm.
It's not even clear from dialogue that the beagle
is alive, let alone Archer.
Suddenly, I'm getting an image of a statue in front of the President Archer Memorial Institute of Theoretical Engineering at Starfleet Academy, or maybe in the lobby—Jonathan Archer as a starship captain, with faithful beagle at his side—and Scotty just "borrowed" him temporarily, thinking old Porthos would be back in place before anyone noticed he'd been missing.
Except wasn't the Scotty's test specific to bio-matter?
Yeah, actually; I was getting a bit carried away with the statue scenario.
In fact, the lead-up line goes "I told him that I could not only beam a grapefruit from one planet to the adjacent planet in the same system ... I could do it with a lifeform." Which kinda suggests he was working with a live beagle. Still, as Wormhole points out above, Admiral Archer's name-check was there to serve as an easter-egg, in the same way as did the order for a Cardassian Sunrise in the shipyard bar, the stenciled "NCC-1701D" on Kirk's escape-pod duffel bag, the tribble on Scotty's work table, and so forth; they were winks to the various other Trek incarnations, rather than hints that Jonathan Archer or a Cardassian or the Ent-D or Cyrano Jones were likely at some point to appear.
Even supposing the alt-verse analog of Archer might still be living in the year 2259, what place would an extremely elderly retired Starfleet admiral and former Federation President have in the story this movie has to tell? There'd have to be some narrative reason for him to appear.