This was actually a thing back in 3.5, in-game you'd see a rotating list of banners for other games. That was when the pager was still attached to Dream Seeker itself too.
And have it say what? "Thanks for playing my game, but if you'll direct your attention over here... and... he's gone" -sadface-
People play more than just one BYOND games anyways so it wouldn't make a difference. It'll actually help people not fly through our game's content like beating half of the game in 1 day just because you avoided sleep. Instead the people might play multiple games a day making them advance through the game's stories a bit slower.
And have it say what? "Thanks for playing my game, but if you'll direct your attention over here... and... he's gone" -sadface-