I might be wrong, but afaik shakycam isn’t actually handheld but rather a steadicam adjusted to get a specific type of shakiness. Since pure handheld footage would have much more high frequency shaking (hand tremor and such) that isn’t the same as the kind of “head moves around” harmonious shaking that you want for a shaky cam scene.
Though it probably depends from project to project.
Professional handheld rigs have a harness that bears resemblance to a steadicam rig, sure (which will making the shaking minimal). And yeah, you could use a steadicam for shaky cam if you wanted, but for the most part that’ll defeat the purpose, and it’s more likely they’d actually go handheld. The degree of shaking is an artistic choice. On a longer lens, you’ll get more shaking I guess, but really when you see something like the Bourne movies, that’s the handheld operator doing some shaking in addition to the natural shake (pure handheld would tend to be less shaky, honestly). JJ has used shaky cam style shooting before, but not on either of his Star Wars films. Practically every shot is either on sticks, a dolly, steadicam, or a crane/jib (as far as I could tell anyway). He’s very specifically gone for an approach with a lot of movement, yes, but specifically a lot of smooth movement.