James,
It basically comes down to weight, I think. Putting a big telescope on an equatorial mount isn’t possible from an engineering point of view. Mirrors that big deform under their own weight when you slew them, which you need to correct for with actuators under the surface. Alt-az is a nice coordinate system, because the deformation varies with altitude, but is independent of azimuth (if you built the telescope properly!).
Another nice feature of an alt-az arrangement is that you can direct the light off sideways to a Nasmyth platform to the side of the telescope. You can even engineer it so the Nasmyth platform never moves. That means you can make the detectors as heavy and unwieldy as you like, and it doesn’t matter. Moreover, you can have multiple Nasmyth platforms with different instruments on, switching between them simply by flipping a mirror which deflects the image off sideways. Most professional telescopes have at least two Nasmyth platforms.
As Andy S says, field rotation is a piece of cake compared to keeping many tonnes of metal aligned to optical precision!
Best wishes,
Dominic.