A Tesla Model S or X draws up to 145 kW of power when rapid charging. It does this for about 15 to 45 minutes depending on battery size and depth of discharge. 

Currently at night, the electric demand on the power grid goes way down and the power plants have to throttle back 60-75% of their generating capability just to keep the grid from having too much voltage on the lines. 

When a fossil fueled generator like a coal plant or gas turbine throttles back, they become less energy efficient and actually pollute more, (per kWh of energy). They also generate less revenue and become less profitable because they are selling less energy but still incurring wear and tear. Having lots of electric vehicles drawing a constant, predictable, steady state power draw from the grid for hours at a time over night would be a welcome boon to all the existing idling power plants. 

Charging at home costs less and is super convenient. I drive about 70 miles a day. I plug in my Chevy Bolt about 2 times a week. It takes 7 seconds to plug it in and about 7 seconds to unplug it. So about 7 x 4 = 28 seconds a week (let’s just call it 30 seconds) is spent actively doing something related to charging up my vehicle. The rest of the time the car is charging is not my concern as I am doing other activities. Compare this to a gas car where you are pretty much a captive audience, standing at the gas pump for 3 minutes a week, (not counting travel time to and from the gas station). 

After a year, charging an EV at home takes 30 seconds x 52 weeks = 1560 seconds or 26 minutes of my time. 

Pumping gas each week for a year takes 3 minutes x 52 weeks = 156 minutes. 

The time saved not pumping gas and instead charging at home is over 2 hours a year. 

A typical charge rate at home is going to be 960-1440 watts from a 120V outlet or 3,600 to 7200 watts on a 240VAC outlet or L2 charging station. Tesla home charging stations can draw upwards of 9600 watts. 

The grid will actually benefit from lots of electric cars charging at night. 

The real risk I believe will come from too many homes charging up vehicles at once on the same distribution transformer. But this risk is actually really small. 

In modern residential neighborhoods 4-5 homes typically share a 50 kVA transformer. Six cars could charge up simultaneously without any issues, (6.6 kW x 6 = 39.6 kW and 7.2 kW x 6 = 43.2 kW) and there would still be margin to run the parasitic loads in each of the homes overnight too. 

With shared scheduling, every home could have up to 6 electric cars a piece (each driving a lot of daily miles) before you would run into problems. 

Back to the problems that next generation's ultra high power 1 million watt charging stations may cause. These stations will utilize large stationary battery storage. Instead of drawing many megawatts of power for minute long surges, they will sip steady state continuously and let the stationary battery handle the brief minutes long surges. 
This will also reduce demand charges, greatly reducing the operating costs associated with providing a charging station.