Short answer: no on cognitive training. They are effective at raising performance on the specific task trained, with negligible transfer to anything else.
To the bigger question of EF, processing speed, etc.: I hear a few aspects of EF that are particularly challenging in your case.
1. initiation (what was referenced as activation). This is, as you correctly note, the skill of getting started.
2. idea generation. Learners on the spectrum often have challenges with this one, as it involves mental flexibility, imaginative and speculative thinking, etc. GT learners and other divergent thinkers also can have problems with this for the opposite reason--which is that there are so many possibilities that they can't figure out how to settle down to one or two. Actually, this ambiguity affects spectrum learners, too, since there's no clear "right" answer. Both ends of the spectrum in idea generation lead directly into difficulties with initiation.
3. planning and organization: Again, this feeds back into initiation. If you don't know where you're going, it can be hard to take the first step.
4. shift/flexibility. You mention this specifically with regard to code-switching.
What does any of this have to do with Coding? Well, yes, it's a fine motor speed/processing speed task, but it also is affected by sustained attention, switching/shifting, working memory, anxiety, etc.
It might help to try a variety of graphic organizers for writing, so that the planning portion is heavily scaffolded. For initiation, sometimes it helps to narrow the choices, sometimes it helps to "prime the pump" with a few examples or sentence stems. Or you can prime the pump by having him start from whatever portion of the paper is easiest for him to write. (There is no rule that says papers have to be written in the same sequence in which you publish them. I write most of my psych evals beginning from the third section.)
It can also be extremely difficult for highly logical/analytical thinkers to understand the function of the smoothing and illustrative parts of writing (transitions, elaborations, examples), as they may be perceived as extraneous fluff. I definitely saw this difference when I moved from an experimental science to a social science. Research publications became (literally) 5 times as long, with much less than 5 times less data. (25 pages on a single descriptive case study, vs 2 very dense pages of reproducible hard data in PNAS.)
Learners who struggle with sustained attention also need personal investment in the topic of writing even more than those who don't, which is why writing about a topic not of his selection is noticeably more difficult. Finding a connection between the assigned topic and something he cares about can make a huge difference in the quality and ease of his writing.
