IMO, the whole "have to take the mountain down to fix the Yeti" is far overstated,
and is just a misinterpretation based on sparse information.
Of course, they can access the "animatronic mechanical parts" as they sit now. Mechanisms with motors need lubrication and other maintenance on a regular basis.
They do replace those heavy propulsion motors on all of the ride mechanisms at WDW due to failure.
(I've been walked out of BTMRR due to a lift-hill complete motor failure (about half-way through, so it affected all ride vehicles.)
A few hours later, the ride was back up and running.)
Back to the Yeti... they could "undress" and disassemble the robot, piece-by-piece to remove it, if need be.
Then, they could do reinforcements and necessary modifications to the mounts and/or the yeti before reinstallation.
My contention is the true reason that they don't do such a major fix is due to the need to shut down the entire attraction for an extended period of time (as has been mentioned) to do this work, as the yeti is so close to the ride tracks.
I'm sure that Disney sees no real plus-side to doing that shut-down and lose the top "thrill attraction" at DAK for so long... when most guests don't miss the yeti to begin-with.
For MYSELF, I'd love to see the giant fuzzball put back in top shape.
Hoping that they do find a good time to do that refurb (as in, NOT while any of us DISer's are on WDW vacation!)