I feel that one of the differences is that a wire is "full" of electrons already, and more are reluctant to enter the wire unless there is an exit path, but as soon as there is a path the current/charge/emf flow occurs all along the wire almost simultaneously.
This is not modelled well with the water mazes/analogy.
It can be modelled, but with a slight twist. If in a water analogy we prefill the maze with a water and then connect it to a source and a drain, then water level will raise near the source and fall near the drain. We'll see then how it propagates through the maze, these waves will meet somewhere, then some more hard to describe action will follow and then the system will settle to a dynamic equilibrium.
This way the water level changes probably can propagate faster then water moves. Though probably to get this result we'll need to make a "deeper" maze. I mean make walls higher so a cross section of a path would have bigger area, so a slow flow multiplied by this area would give us a big volume of a water moving. Big enough to raise level of a water fast in a whole section of the maze.
The movement of water molecules is typically much slower than the wave speed of water.
When you turn on flow to a filled pipe, the water is reluctant to enter the pipe faster than other water is leaving the opposite end (mediated by pressure waves).
Mainly because the maze is empty to start with, and the water must flow simultaneously to all points equidistant from the entry until the exit is "found".
The analogy would work better if the maze would start with entrance and exit closed, and filled with water to half height. IIRC the author mentioned this in a comment under the video.
At one point in the video, he appears to splash clear water into the filled maze. You can see the non-moving parts lighten their colour compared to the active path.
This is not modelled well with the water mazes/analogy.