Woodworking Q: Do I need a router or will a Dremel routing bit work as well?

This.

Just build it without overhang. Seems like a strange design that requires routering off an edge. It makes it sound like they threw a bunch of raw lumber in a box and you’re suppose to make nice with it.

And FYI, if you don’t want saggy shelves then screw or glue strips of wood to the edges to create an I-beam effect.