Even the use of 'flow' is misleading. It's barely trickling through the wire...
The movement of electrons is inconsequential. It’s the magnetic fields between electrons that provide the power in an electric circuit. These fields actually don’t even travel through the wires! They move around the space outside the wires. Ask anyone who has routed a differential signal pair :)
Veritasium has a video that goes over this. It was pretty controversial at release but it gives a good overview. https://youtu.be/oI_X2cMHNe0
And why magnets work due to relativity https://youtu.be/1TKSfAkWWN0
You can see the voltage “ripple” through both branches of the circuit, reflect of the ends of the branches, and eventually settle into steady state. There are limits to this way of thinking (for example, water can’t travel in the space between “tubes”, but electricity can), but it is a very good way to approximate it.