Why can't the neutral and live wires be exchanged when wiring a plug?
There are two fundamental requirements for safety unless the design of a device guarantees that no electrified portions will have a path to ground other than via the neutral:
The hot wire needs to be fused unless the wire can pass 20 amps continuously.
If the hot wire is disconnected, the neutral wire must be disconnected as well.
If a device didn't have a fuse on the hot wire, nothing would control the over-current in the event something shorted to ground. If the device fused both sides and--under a different failure condition--the neutral-side fuse blew before the hot fuse, the device could appear to be inert while its innards were all connected to the hot wire.
There are two says a device could be constructed to be safe without a polarized plug:
Design it so that there is no plausible accidental current path anywhere except between the hot and neutral (whichever is which).
Use a double-pole fuse which will open both sides under an over-current condition. While it's possible to construct fuses in such a fashion, such designs are much more complicated and expensive than fuses that merely need to open the wire carrying the excess current.
Using polarized plugs is often easier than doing either of the above.
Because the electrical code (at least in US) requires neutral and ground to be connected, usually at the service panel as shown below. While simple systems might not care which input to the load is live, many systems will have some non-trivial connection between their neutral input and ground.
Note, also that fuses will be more effective if they are located on the live wire than on the neutral.