There's quite a bit of perhaps inadvertently misleading information here.
Plugs and sockets overheat due to power dissipation at or near the connection. This is nearly always because of high resistance at the connection due to looseness, corrosion, improper or broken attachment of wires to the connectors, or something along those lines. The power dissipation is of course related to the current that's flowing, but a properly seated connection between connectors in good shape is more than able to carry the rated current (30A in this case). If more current were to flow for more than a very brief time, the circuit breaker would trip to protect the circuit.
Low voltage causes many devices to use less current (all resistive devices) and a few to consume more current (switching power supplies, some electric motors in certain applications). But the voltage itself is not the cause of the connector failing.
Similarly, undersized wires are dangerous when used with more current than they can carry (which is what being undersized means), but they do not cause the connector to melt. The wire itself may get hot and start a fire or otherwise be dangerous, but it wouldn't cause the connector to fail in itself. That's a moot point here as the wires involved were apparently all quite adequate for the current.
Note that power dissipation of a resistive connection increases with the square of current. A one ohm connection, which is a very poor one, would drop ten volts at ten amps and need to dissipate a hundred watts; encased in plastic as connectors are, with little chance to dissipate heat, it would melt in short order. A connection with a resistance of one tenth of an ohm, at ten amps, would dissipate ten watts; that would be plenty to get mighty hot internally over time.