Not a question of advantages, a question of tolerances. At the track size used now, thermal expansion and process tolerances are enough that even with a position specified to the micron from the center, you'll have different track numbers between platters solely by variance. This is why every track has MANY markers on it identifying its track number, and their values are used as feedback for the servo mechanism that moves the head. When seeking to track X, it will pick up markers while it moves and after it stops moving, use them as feedback, and continue to move in the proper direction. Nobody knows how many microns from center track 1234567 is, nor cares. It is more or less a PID controller, with desired value being desired track and current value being current track seen under the read head.

I would never have guessed how much thermal expansion had to be compensated for in modern hard drives, I would have assumed they were built such that no degree of precision that required accomodating changes in physical parameters was attempted, and/or that materials science had found an ideal set of alloys for the case, the platters, the magnets, arms etc, that all behave in a predictable way under a given thermal map, so that addressing the surface was nearly always a deterministic process.

Have there been attempts to use active cooling/heating (i.e. thermal sensors combined with piezoelectric effect elements embedded in the housing) to maintain a precise, consistent reference internal operating temperature, as a means of eliminating the need to accomodate thermal-induced drift in tolerances?

why? making a feedback-driven servo is a well understood task. it is not hard. and even if you knew the precise micrometer position to move the head to, magically compensated for ALL variables, you'd still need a PID controller to MOVE you there, and feedback to help you measure where you are. You'll have added a significant material science problem, but solved nothing

Sorry if I'm coming across as presumptuous or something, it's just surprising to find that modern hard drives are still using the same basic mechanism that I saw inside the washing machine-size drives my dad worked on 40 something years ago, that would almost walk themselves out of their mounts thrashing their giant head servos around.

What you and others are describing is strikes me the way that opening up the package of a modern memory chip and finding a fabric of millions of tiny magnetic cores would, rather than a silicon array of capacitor/transistor cells - that for some reason no fundamental rethinking of the problem has prevailed and thus the only developments in RAM for almost half a century is miniaturization of what was being done in the 1960's.

The first PC HDDs used stepper motors and absolute positioning with cylinder geometry, just like floppy drives. Then they moved to voice coils and remote encoders (e.g. early Quantum drives), dedicated servo surface (drives with this report an odd number of heads), and finally embedded servo with non-cylinder-based layouts where each surface can have different track pitches and sectors per track.

I wouldn't call that "the same basic mechanism" as there have been lots of refinement over the years, but it turns out that having the heads themselves find what they need to read was the best solution.

AFAIK all optical discs also use a similar scheme.

Incidentally this is why a HDD with a bad head or media area will make clicking sounds, sometimes very loudly, as the head actuator slams against the stops since the controller can't see the signals it's looking for and can only sweep the surface in search of them.