How is energy transported at the speed of light if electrons don't flow like very fast water?
Let me reply with the hydraulic analogy, i.e. with replacing electrical current by water flow.
Open the water tap in your kitchen. Then water comes out instantaneously, although the waterworks feeding the water pipes might be many miles away from your house.
Of course, this is not surprising. Before you opened the water tap, the water was already present in the pipes all the way from the waterworks to the water tap in your kitchen. It is the pressure, not the water, which propagates so fast (theoretically with the speed of sound) through the pipes.
This water scenario above is very much analogous to the electrical scenario.
When you switch on the light in your room, the electrical current through your lamp begins to flow instantaneously, although the electrical power station might be hundreds of kilometersmiles away from your house.
This is not surprising here, too. The electrons were already present in the wires all the way from from the electrical power station to the switch and the lamp in your house. It is the voltage and the electric in the wires and the electromagnetic field around the wires, not the electrons, which propagatespropagate so fast (theoretically with the speed of light) through the wires.