So basically what you're proposing here rests on the assumption that classic WoW is going to be a short fad that will decline very soon. I disagree with that, as I'm personally very fond of replaying old games over and over. Nostalgia tends to come back, you know. However, while it's easy to replay a single-player game (you just find your copy of the game, reinstall it if needed, and have fun all over again), you can't do that with an MMORPG (and any other online game, for that matter). Once you update an online game, the old version is gone forever. So what you are proposing would go against the very idea behind classic WoW's relaunch.
Even if the interaction between the classic and current content servers is going to be very limited (say, cross-realm arenas), how exactly are you going to do that? Remember that even basic combat mechanics are different between classic WoW and current content (Battle for Azeroth). There are no weapon skills since Cataclysm, the hit mechanics are very different, the buff/debuff limit in Classic is smaller, to name just a few things. How exactly are you going to make this work together, and, more importantly, how are you going to balance all that?
However, recording replays for dungeon and raid runs and for battleground/arena battles is an interesting idea.