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.