What if you double framed it somehow?
I just cannot believe that with how "magical" movies are there is no possible way one could make 4:3 contact to 16:9. It might be nigh impossible, costly and massively time consuming, but there has to be some way.
Anything's possible in this day and age. The question is whether it's practical. The process of creating new sides you're proposing is essentially the same as the 3D post-conversion process, which paints in new background details behind foreground elements to simulate a second viewpoint, a painstaking, brain-numbing process requiring hundreds of artists and months of work. Some quick research says that at its peak, just twenty-five movies a year are post-converted, which would only cover about 2/3rds of the run of TOS alone if the entire Hollywood apparatus was devoted to it. It would actually be more work, since you'd be inventing two large continuous stretches of frame, rather than just filling in gaps around the edges, including all-new characters, so it would take years of full-time work from an entire industry.
And as
@Noname Given points out, it'd just give you a weird, center-justified shot compositions for every episode. Though if you're already painting in so much extra material, why not just cut-and-paste the characters and recompose every shot for widescreen? We'll have all of Hollywood working on it anyway, let's make it a new five-year-mission. I actually don't mind, I think post-converted 3D is terrible, so it'd be nice to keep those companies busy on something else for a while.
Honestly, as suggested way back at the beginning of the thread, it probably would've been fine to just have a third seamless branching option with the widescreen effects shots. "The Dark Knight" is an example suggested with it's IMAX transitions on home video, but "Star Trek Into Darkness" is an even better example, and not because it's Trek. Aside from two or three complete action sequences filmed in IMAX (and so displayed full-frame 16x9 on the IMAX-ified Blu Ray), every pure-CG shot in the film was done at IMAX framing and resolution, so the framing is constantly switching back and forth with every space shot, sometimes alternating shot-by-shot in action sequences like the space battle and crash sequences near the end of the film. I'm sure that at the time the TOS-R Blu-Rays were released, it was conventional wisdom that switching aspect ratios would be distracting (and it probably still is, since those are the only two films I know that do it on home video, and then only on specialty releases.* The Force Awakens also had taller IMAX shots, and I don't believe that version has been released on home video. Maybe they're saving it for 4K), but I think it's fine, at least for exteriors and interiors. Maybe if the interior shots were switching back and forth, it wouldn't work for me.
*Not counting movies that change aspect ratios for artistic rather than technical reasons, like The Grand Budapest Hotel.