At the end of the day, the fact remains that the bridge of the TOS ship and the bridge of the ST2 one are about as similar as an onion and an oxcart. No amount of "hazy recollection" would make Picard identify Scotty's simulation with a refitted bridge.
could have been resting unrefitted in the Fleet Museum since the 2250s for all we know - say, a vessel disabled early on in the lifespan of the class and patched together enough to serve as an attraction. Scotty doesn't comment on whether he's familiar with the museum ship or not; he just wants to know how familiar Picard is with this ship type (the unrefitted one), and accepts Picard's statement that the museum ship helps the 24th century captain a bit in that respect.
Whether Starfleet refitted old Constitutions
en masse or not, we don't know, since we don't see ships other than E-nil and E-A (plus a corner of a further vessel at the end of ST4, but that could "in reality" be of just about any class). We're free to speculate that all ships save for a few hopeless cases were refitted and scores also built to the new format (like many fan sources do), or that only two ships were ever refitted, both for purely experimental or otherwise special purposes. Any and all explanations allow for an unrefitted or a refitted specimen to be sent to the Museum, but clearly only one or the other, as per "Relics". And since Picard does recognize Scotty's simulation from visual cues that have absolutely nothing to do with the refit (and no, he can't be doing that on the basis of dedication plaques or graphics or whatnot since those aren't visible to him from where he stands - Scotty blocks the view to the distant, small-print plaque, and the only graphic that could help Picard is on that side of the turbolift alcove that is hidden from him), it seems obvious that an original configuration vessel is stored there.
...Unless Picard utterly failed to recognize the bridge, and merely hazarded a guess, considering what he knew about Scotty.