your chemistry and physics aren't quite right. neither are your numbers for coal's share of electricity in the US (it's roughly 30% and declining). but you are absolutely correct that electric cars are far from a panacea and that electrification generally isn't all that great.
let's start with gasoline. chemists often call it BTEX. that's benzene, toluene, ethylbenzene, and xylenes. none of those really resemble methane except for the basic building blocks (hydrogen and carbon). they're all aromatic hydrocarbons, which have a ratio of carbon to hydrogen that's closer to 1:1 (1:1, 7:8, 4:5, 4:5, respectively) than the 1:4 of methane. I don't know much about the chemistry of coal, but it does have higher ratios of carbon to other elements. coal's energy density is also greater than BTEX, which complicates comparisons.
internal combustion engines are very inefficient. they're heat engines: the heat produced by oxidizing fuel does pressure volume work as it pushes a piston to turn a crankshaft (unless it's a rotary engine, which are pretty rare, or a turbine engine, which are even rarer). a large majority of the heat produced, however, is wasted. some of it through the tailpipe, some of it through the radiator, some of it through the walls of the motor.
electrical power plants are much more efficient. a far greater portion of the heat energy is used to pressurize the working fluid and turn turbines. friction losses in modern turbines are minuscule compared to the inefficiencies of an ICE, as are the inevitable inefficiency of transforming chemical energy to kinetic then electrical energy. transmission losses are also certainly an issue. again, all of these losses are dwarfed by the inefficiencies of an ICE.
instead of fossil fuel, let's say the electricity came from hydro-electric generation. there are a few notable exceptions, but dams generally involve what amounts to the wholesale destruction of otherwise hugely complex and productive ecosystems, not to mention that they cause substantial releases of methane gas from sediment in reservoirs. methane is shorter-lived in the atmosphere than carbon dioxide, but very much more potent as a greenhouse gas.
there are other sources of electricity, but honestly, I think the bigger issue is the ubiquity of automobiles generally, our habit of arranging our entire built environment around them for the last hundred years or so, and the un-examined assumptions that we can't do without them and that there must be some way to keep using them everywhere but without the negative consequences (I'm speaking primarily about the US. other countries aren't necessarily blameless here, either, but the US is the worst I've experienced). once the need for and consequences of so many automobiles are honestly considered, it becomes obvious that their overuse is incredibly damaging in so many ways. that isn't to say they aren't a very useful tool. they most certainly are. but they are overused to an absurd degree. if we collectively decided to only use automobiles when they were actually the best tool for the job (and designed them better for how they're actually used), I don't think it would matter so much how they were fueled, so long as care was taken to avoid toxic pollutants.
which brings me to diesel. compression-ignition engines are, as you suggest, more energy efficient than spark-ignition engines. they also last a hell of a lot longer. a huge drawback, however, is that increasing efficiency comes with increasing air pollution in the form of nitrogen oxides. it's obviously more complicated than this, but increasing efficiency in a diesel engine involves increasing temperatures. that, in turn, leads to more creation of nitrogen oxides which have very negative impacts on air quality. there are urea injection systems in many modern diesels that address this, but it's still a problem. and it's why many European cities are enacting laws that will phase out and eventually ban diesel engines.
there again, though, if diesel motors were used far more sparingly, impacts to air quality wouldn't disappear, but the impacts to environmental quality and public health would be far more acceptable.
in the end, using a 1000-lb contraption to move a couple of 200-lb humans around is more than a little ridiculous, regardless of how that contraption is powered. using a similar 1000-lb contraption to move 5000 lbs of cargo is rather less ridiculous. and I'm being generous here. in reality, many (if not most) private automobiles are much larger than 1000-lbs and are only moving one human.
I've gotten fairly far afield from your question, so I apologize. the answer to that question is that, yes, powering cars with electricity is better in many ways than powering them with internal combustion engines, especially if you're primarily concerned with carbon emissions. but in many other ways, they're essentially the same. in a few ways, they're worse (mining of lithium and other elements for batteries).