Simple version:
Yes, a lightbulb will work no matter which way the electricity flows.
But… what if the bulb is removed from the socket?
Now the switch can prevent the lead from touching your finger, or you can have the wiring reversed and the “ground” is always hot because there’s no switch on that side.
The polarized plug is purely for safety so that the switch actually does its job. And this expands for more complicated electronics as well.
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