Executive function is a frontal lobe thing
It does not fit well within the "left vs right" brain paradigm as far as I can tell.
We have tried many different approaches and so far none have really "worked" any better than any other. And there are some things like math facts that just cannot be done any way but memorization.
Understanding, remembering and following directions/steps is a complex skill that can fail for many different reasons. EF issues is just one possible area of difficulty that may result in problems following multiple step directions. In my daughter's case the problem is auditory processing and extremely slow processing speed. These could be seen as partially EF problems but really have more to do with other aspects of cognitive function.
EF problems are difficulties with attention, control and metacognition - switching attention when you want to or need to, focusing attention for extended periods regardless of the novelty of the stimulus, controlling emotional reactions to situations, creating and maintaining order at the informational and physical level, having insight into your own mental processes. These skills develop along with the frontal lobe myelination and development as children grow up. That's why problems with EF may not become apparent until middle school when most neurotypical children's frontal lobes have developed to the point that they can do these things easily and reliably.
