I remember the Animaniacs and the Warner Brothers next-generation cartoons, Tiny Toons. I had a 6 year old daughter at the time. I watched those with her, and thought they were outstanding. The Animaniacs were cut from whole cloth, not a reboot. And they were the best. But the Warner Brothers Tiny Toons cartoons were almost as good.
I think what makes the difference is (1) really good source material and (2) excellent execution. The connection of Steven Spielberg to the project made a big, big difference.
I believe the popularity of the animated cartoon series with "kid versions" of some of the Muppets ("Muppet Babies") led to this fad.
I remember the Animaniacs and the Warner Brothers next-generation cartoons, Tiny Toons.
^The segment of "Animaniacs" you had referenced actually had Yakko singing about all of the countries of the world, not Wakko.
Muppet Babies was both on CBS Saturday Mornings and in syndication (it was aired on WPIX and many many other huge independent/Fox stations in the late 80s and early 90s). Ah, the times where kids could watch their local independent after school for some cartoons!