But they do keep a Constitution as a museum piece - and she's not the E-A, as her bridge apparently is of the late 2260s standard still. Also, she's the only one.
Unless I'm reading too much into "Relics", that is. Picard waltzes in on Scotty's simulation and recognizes it for Constitution class, which is fine, because we never saw the TOS design associated with the bridge of any other class. And then he says there's a Constitution class in the museum. Doesn't mean Picard recognized the simulation because it was identical to the museum piece - he could have recognized it because visuals from the 2260s adventures of various Constitutions were part of the public consciousness for some reason or another.
Picard may have "cheated" a bit and either looked at the program before entering the holodock or drawing on knowledge of Scotty's career. His expression as he looks around seems like is isn't trying to identify the ship class so much as he's grappling with how to approach his conversation with Scotty and is just buying himself a few moments and testing the waters. Or at least that is how I read that scene.
After all, I don't think Picard was there by accident. He knew Scotty was going through a rough adjustment, as anyone in his position would, and was trying to feel out how Scotty was feeling. That Picard walked in on Scotty's holodeck program instead of wating for another time to visit suggests that he already suspected Scotty was having difficulty, possibly having made discrete inquiries with Data or Laforge.
The fact that Scotty was running a holodeck simulation of his old ship probably spoke volumes to Picard about Scotty's current emotional state. From the moment Picard walks into the holodeck, he knew full well it was the bridge of the Enterprise, which is Constution Class Ship, and the one Scotty's served on during the best part of his career. Anything he said and did was simply "blowing smoke" as he until he finds a way and a good moment to ask the to the question he REALLY wanted to ask, "What do you think about the Enterprise D", meaning "How are you REALLY feeling here." Point is, there were a lot of things going on with this scene, making it not a good one to glean how recognizable and distinctive the appearance of the bridge may be to that class.