EVs offer some definite advantages, and lots of advantages that depend on the specifics.
short answer: yes, it's better than driving a fossil-fuel car.
Long answer: it may seem to make less sense if you only consider a very small range of impacts, where all of the costs are visible, but only some of the benefits. Only when you zoom out to the long-term system level do you capture all of the benefits. And there are even better alternatives.
In terms of their quantified advantages over petrol (gasoline) & diesel, it will depend on just how your electricity is generated, and what the full lifecycle costs are of each vehicle.
carbon in motion
Even if the extra electricity is provided by coal, an electric vehicle is so much more efficient than a fossil car, that there will be carbon savings: electric cars are typically 80-90% efficient, compared to about 20% for a fossil car. So even with a coal plant at 33% efficiency, there's still a carbon saving from the electric.
They definitely reduce local pollution - unlike fossil cars, they don't emit NOx, CO2, CO or particulates. And they are much easier to decarbonise than fossil cars - they just need renewable electricity to recharge.
As I wrote over on skeptics.SE:
Electric cars are about 4× as efficient as fossil-fuelled combustion engines, tank-to-wheel: ICE engine efficiency is around 20%. Electric engines tend to be around 80-90%. For the whole-cycle energy-efficiency, then well-to-wheel would be the efficiency you were after, and then it would be very sensitive to how your electricity would be generated.
Note that pretty much all electric cars benefit from regenerative braking, and very few fossil-cars do.
Efficiencies do depend on the drive cycle: and whereas ICEs tend to be optimised for speeds around 85km/h, the efficiency of electric cars decreases with increasing speed, just as the core physics would lead you to expect:
For more information on "power-station to wheel" efficiencies of electric vehicles, see the US Gov Fuel Economy site.
The chassis of an electric vehicle is usually lighter, too (except the Rolls Royce Electric Phantom), so there can be lower impact on materials for the chassis. The batteries are materials-intensive; they are currently so valuable that a lot of attention is paid to end-of-life reuse and recycling of the materials.
Electric vehicles can help with the market penetration of exogenously-variable renewables such as wind and PV. They release some of the stranglehold that oil companies have on the economy and politics. They make for quieter streets, and quieter car journeys. They eliminate the local pollution from car exhausts (tail pipes).
So in general, yes, electric cars are a significant improvement over their fossil-fuel guzzling competitors. I've seen no evidence that the batteries end up in landfill after just 3 years of life - that seems to be a bizarre and unsupported claim. Such a practice would not be remotely sustainable. I have seen evidence that some manufactures are seeking to recover the batteries: presently, at least here in the UK, a lot of electric-car suppliers are leasing, rather than selling, the batteries, which means that they'll go back to the manufacturer at the end of their life for complete recycling. After all, a dead battery contains all the ingredients needed to make a brand-new battery - the thing that needs doing, is reversing all the decay chemistry, which is just a question of energy inputs.
Patterns both of car ownership and use have changed hugely in the last 4 decades. And the change in the 4 decades before that was huge too. 4 decades is enough for really big changes in transport patterns. For several reasons. It's 1.5-2 generations, giving plenty of time for cultural attitudes to shift. It's long enough for disruptive technological innovations to gain big market share. And it's long enough for a sufficient amount of land-use to be reshaped around new transport patterns.
Current car ownership doesn't make a lot of sense, from lots of perspectives. Most cars spend almost all their time stationary. Quite a lot of dwellings have different cars for different purposes. Most cars are able to drive hundreds of miles on a single refuelling, but very rarely if ever do so.
It's possible we will see completely different patterns of ownership in the future, with various different forms of shared ownership.
It's also possible that the growing obesity epidemic in parts of the developed world, the internalising of negative externalities into energy prices, and a better public understanding of the prerequisites for and importance of a vibrant public realm, lead to a resurgence in walking and cycling, negating the need for a big proportion of current car ownership and use.