I'm right handed and it seems like my left hand is "weaker" or less capable than my right hand. Isn't there any methodology to correct this problem that could be called a left hand methodology? Or is that solved by focusing equally on the right hand and the left hand? It seems to me like I should be focusing on my left hand more, since it is not as good as my right hand? 
Yeah, well, join the club!
I wouldn't be so presumptuous as to call anything I do "methodical," but while I do a lot of fundamental work with the good old scales (including some odd ones) at the octave, the third, and the sixth, I also work the LH specifically on things like the G major prélude of Chopin, many passages in Bach and Beethoven which feature "running" figures in the LH, and, really, just anything I can think of that helps me move closer to a more complete fluency. Yes, even Czerny exercises: some of them are kind amusing to me, even if some of them are more or less cribbed from ideas in Beethoven.
I happen to have a strong LH for many different ideas, probably thanks to background in stride piano, boogie/rock and roll, New Orleans music, walking the bass on the Hammond organ (or piano, very occasionally), and all kinds of arpeggios, but the above paragraph is just how I've been trying to improve. Being so used to doing chords and wide leaps if, anything, sort of perverted the way I think of the LH, so there's a good bit of neural reprogramming I've had to do.
I have found for a different kind of dexterity the vast majority of Bach I've read to be helpful, even if it's only obliquely related to doing strictly scalar works. It's firmed up an amount of control, if nothing else.
Not that I can't pretty play fast scalar passages in the LH when needed, sort of, but it's just not as automatic and fluent as I'd like it to be, in order to execute improvised ideas or to cut down on time spent on repertoire that uses the LH more completely and melodically (even if the "melody" is rudimentary and based on scalar motions).
So, that's not much of an answer, since it just boils down to "practice," but that's my method.
But, maybe the one thing that could be helpful is that I have some very concrete reasons for training in this manner, not really just because I think I should, in abstract, idealized sense.
Listeners are going to observe the singer more so than the pianist because that is where the melody lies, sure the pianist can emphasise and decorate the melody and take more attention when the singer has a break but they don't want to over play and drown out the singer with melodic lines while playing with them.
You make some critical points in your post. I just thought I'd share my annoyance at trying to accompany less-than-accomplished singers at pop/jazz music: many times it seems they get annoyed if I'm not actually playing the melody along with them.
Which is, IMHO, ridiculous: it robs the music of any hope of expression and coherence. Not that there isn't a place when one might one to double the melody, in, say, an instrumental arrangement, perhaps by providing a harmonized version of the melody.
Just a pet peeve of mine and why I really try to make myself absent if various people, at, say a party of some kind, make rumbling noises about wanting to sing a tune. Yet another reason to skip Christmas gatherings, as if the execrable music weren't reason enough. Vince Guaraldi excepted, of course!