The equatorial mount is an analog computer that performs the transformation from local terrestrial coordinates to celestial coordinates. Building it into the structure of the telescope is necessary for the analog approach: while you can make a smaller model, there is no analog technology that can transfer the result to a larger structure with the required accuracy.
So, the obvious innovations are the digital computer along with digital angle encoders to allow it to tell where the telescope is pointing. Sounds simple, but getting computers to operate reliably on mountaintops required a fair amount of finesse: shaky power, lightning-induced currents, ESD, temperature fluctuations and other hazards (Mauna Kea ash is nastier than ordinary dirt) needed addressing. The mount, of course, was not the only thing people were interested in automating.
The question notes that “the very largest and heaviest reflecting telescopes equatorial mounts are massive and unwieldily and require huge counterbalances compared to alt-az mounts which can have the azimuth bearing right on the ground and the altitude bearing straight through the telescope's center of mass.” This is true, but it also reflects old misunderstandings.
The first big alt-az telescope was the Soviet BTA-6. It indeed mitigates the structural problems associated with balancing its massive mirror. It is, however, a mediocre telescope, plagued by poor "seeing".
The next one was Smithsonians's MMT. In its orginal form, it used six lightweight mirrors supported by a lightweight structure in a boxy building of minimal size. The whole building rotates in azimuth, while the telescope within rotates in elevation.
Unlike the BTA-6, the MMT was a very good telescope. It had remarkably good seeing. The contrast here changed the way that astronomers thought about telescope architecture.
Why the difference? The BTA-6 has enormous thermal inertia in its heavy mirror, heavy structure, and oversized dome. That means that it never really comes close to thermal equilibrium with the air around it. It's always surrounded by turbulent convection currents. Atmospheric turbulence blurs images, an effect astronomers call "seeing". On the other hand, the lightweight MMT operates much closer to thermal equilibrium, so there is less turbulent convection.
Before the MMT, seeing was thought to be mainly a property of the site and not the telescope. The MMT demonstrated that for big telescopes, much of the seeing originated in the telescope and observatory structure.
So, it wasn't simply that technical innovation enabled alt-az mounts for huge telescopes. It was necessary to find the correct motivation to use them. As it turned out, dealing with huge mass was not good motivation. Alt-az mounts became part of a strategy to make lighter telescopes with better thermal properties.