I disagree it doesn't do that, Babette would return to the sanctuary and find it destroyed and everyone dead she'd move on, likely leaving Skyrim all together. She wouldn't know about the sanctuary in Dawnstar let alone the password to get inside. Cicero would return to the Dawnstar sanctuary with the Nightmother once he learns Astrid's chapter has been exterminated. The next game will really need to decide which path is canon eithe the Emperor is assassinated or the Dark Brotherhood is effectively destroyed.
1. Yes, it DOES do that.
2. How could a vampire, who has been killing people for centuries, possibly move on?
3. The next game will likely have both; The Emperor is assassinated AND the Dark Brotherhood is effectively destroyed.
4. Since Cicero can die, it's entirely possible that the Dark Brotherhood dies with him, leaving the Night Mother stranded somewhere for a new Listener to pass by and rebuild the DB. Bethesda loves to mess with our heads and never give us concrete answers. For example, what happened to the Nerevarine? It's rumoured he/she went to Akavir, but it's just that; a rumour. Could it have happened? Possibly, but it's never confirmed. Champion of Cyrodiil? May or may not be Sheogorath. Dovahkiin? No idea yet, but the answer will be deliberately cryptic. Since the DB questline is not a main one (which likely means no Dragon Break) yet there are constants between the two, it very likely means that the canon response/development will be that the DB is still on the rocks/not as powerful as they were, and the Emperor was killed by a lone, skilled assassin; whether or not he/she was a DB assassin will likely never be known, but could be implied.
Bethesda loves to mess with us this way. We even see it other faction questlines. If the hero doesn't do these things, some other mysterious person does. For example, my main in Oblivion never touched the DB quest, but it is still official canon that there was a purging of a DB Sanctuary in Cyrodiil in 3E 433.