Hopefully, this will lead to a bit less violence over fuels.
In other words: No, the scenario you describe won't happen.
Solar panels seem to keep getting cheaper; distribution via the power grid does not. This may lead to property owners with enough land eventually finding an economic incentive to get most of their power from solar panels on their land instead of using the grid.
With electricity, it doesn't matter what the source is. If your electricity comes from wind, solar, hydro, geothermal, nuclear, whatever.. You can generate it where you are. Or you can use local coal/gas/biomass/biogas resources. But electric cars can be run on whatever, while gasoline cars can only run on oil (first generation biofuels used now are now very energy positive, if at all, so they don't count).
I guess a good real-world example of what I'm saying is: Right now, we're not really seeing big crises and wars about electricity, and I expect that it will continue in a similar fashion in the future, with or without EVs on the grid.