Out of the kid shows I have watched I only really remember catdog, beetlejuice.
If you remember Beetlejuice, that means you're probably a 90s kid. That would mean other kids shows from that time include Rugrats, Ahh Real Monsters, Rocko's Modern Life, Ren and Stimpy, Gummy Bears, Tail Spin, Bonkers, and Animaniacs...
Huhh... I suppose none of those shows really DID have the storyline specified at the top of this thread.
Anyway, that's the cartoon shows. For live action, or live action + puppets, we had Thomas the Tank Engine, Mr. Rodgers' Neighborhood, Eureka's Castle, Fraggle Rock, and on the pure live-action there was Boy Meets World, Clarissa Explains it All, Kenen and Kell, and Sabrina the Teenage Witch.
Don't think that storyline appeared in any of those ones either. And yet, I'm also a 90s kid (as evidenced by my ability to rattle off all those titles) but I do very much remember seeing the storyline specified at the top somewhere.