Having upgraded n-years ago to the Logitech M570--somewhat programmable (don't use) and with the most useful feature, the scroll-wheel also incorporates [that missing Button #2 in the middle!] I had presumed this as close-enough-to Perfection. It had managed to slip off the table a couple times, seemingly undamaged. Seemingly. Finally, the middle button: the essential Instant switch to iStat (close as we get to HAL-9001) for OS-X health status, just after Activity Mon is peeped.

So off to eBay; found a New/sealed one for ~$32 deliverd ... Fast. In scanning some reviews found a complaint re quality-of-recent models of Logi trackballs.. Ths guy must be mousing 12/24 forever; his advice: replace every 1~3 years depending. [On this advice, too: never get a refurb of a T-ball] I haven't found the trick to fully-open the broken sucker ..stlll hidden screw? to doverai-ni-proverai his complaints, but will. tl;dr at a mere $10/year if he's right: you can't pry this sucker from moi hands except OMDB.

The new one reminds of the eventual wear-down of the silky-smooth plastic 3-points supporting ball [Basics: "3-points determine a plane"]; so much for the redundant-constraint of all chairs except tripods. Found that the middle-key had other effects in broken one--for many months making 'iffy that switch. New one--the ball floats as a massless/frictionless impossibility. Moi happy. So there you have it: replace your own, periodically (those of us'ns not still using those carpal-tunnel of-the-shoulder antediluvian-Mice things)

Will also throw a small cloth over the sucker when off--noted already that dust enters the 3-switch area and old unit needed ball extraction and cleaning of the exposed pivots regularly + hand lotion(!) sparingly: because of wear on the tips. Sic Transit.

Have yet to determine what is resisting uncasing; if successful, your tip may cause me to do a bug-hunt for my magic elixir which deposits a functioning Ag coating, though unlikely to try.. the drops + wear on the crucial 3-points makes this a basket case. Can 'fix' anything techo, pretty much: but my Time is worth more than saving $10/yr: it likely goes to Recycle bin.

(At least--have already trained-self: always use butterfly-like force on that frictionless-ball);
save the Fucking-points! a bit longer :-)

While my particular thumb can move that sucker fast--setting Acceleration just-right--to a Point-X, I can see that there's little thumb-memory of the sort you describe; I'd have also to eye-radar to X, in a trice, (another step). Tested: if you put some small squares or circles on a blank page, have to hit some sequence of these to do chores such as you describe.. the ∑-delays are indeed Additive. Rats!

IT-tragedy: a theory killed with a fact. (Though, unless disabled-thumb-workings), the savings of shoulders and such does seem an authentic factor (for those not forever dwelling in Mudbrick Tech Inc.) Solution: have both, eh? Have seen work-spaces so equipped. Dunno their reasoning ... unless ambidextrous? ..Hmmm: wins contest.

Added to listicle: Nothing is ever 'simple'.