Relevant to the OP's example, "Meeskite" was sung by Herr Schultz in the original production of Cabaret but hasn't been seen in a major production for decades. I sang it when I played Herr Schultz in college, and I thought it was a fun song and an effective way to alert Ernst Ludwig to the fact that Schultz is Jewish, but I accept the fact that the creators were willing to let it go in the interest of a leaner show.
What production of CABARET did YOU see?! The stage version of CABARET (regardless of which revival) has several songs performed outside of the Kit Kat Klub (Frau Schneider and Herr Schultz anyone?). As mentioned above, Bob Fosse’s 1972 film adaptation eliminated and added characters and only kept and added songs solely performed inside the Kit Kat Klub. Only “Tomorrow Belongs to Me” is performed at an outdoor park in the film.
I love "Why Should I Wake Up" and don't think it's thematically disposable. Would also help Cliff actually register as a character in a way he hasn't in productions I've seen.