What's the purpose of a player in a doomed timeline sleeping when the timeline resets and transferring their memories to the alpha?

This is hard to tell, because it’s unclear whether this is a Mechanic of the Game, or if this is merely because Rose is a Seer- We see Terezi also gathering memories from the GO! Doomed Timeline after all.

Either way, it seems that when the last Observer of a Timeline disappears- That means, everyone’s dead or gone except one person, and this one person ‘gives up’ or ‘goes to sleep’, the Timeline ceases to exist. This is the only clear vision we get on how Doomed Timelines just. Stop. And while not specially necessary or super useful, Rose used this method to give her Alpha self all the knowledge she gathered through the few Months they were in the Doomed Session, tell her to get to Dave on Derse early, wake her up early as well, and potentially giving her some insight on where to go next- Contacts with the Horrorterrors, Knowledge of the Game…

Dave came back from the Doomed Timeline with items and as a Sprite Guide for Dave. Rose sent her Information back, instead. Fitting for a Knight and a Seer.