The power grid hasn't melted even though the plugs at the parking spots are on a timer and usually run two hours from 0600 to 0800 so the car is nice and unfrozen when it's time for the daily commute.
Now when we are moving to PHEVs and EVs some people are _very_ worried that an EV using the exact same plug with the exact same load steadily from evening to morning will melt the entire power grid and the local wires and melt the plug. Dunno why.
Street parking is still an issue, but there are solutions for that too. The power could be pulled from street lamps for example.