Again, yes, but to clarify, the 60+ Hz I was referring to was not the servo loop update rate, but the rate at which (loosely speaking) the servo loop setpoint is updated. (1)

For example, KUKA robot arms can operate in a mode where a motion path is planned and a sensor on a tool tip can make slight adjustments to the motion path on the fly. The points on those motion paths (as well as corrections) are updated every 4 or 12ms (83.3 to 250 Hz).

My point is that servo motor drives do indeed implement (sophisticated) PID controllers for current, position, velocity control loops. (2)

Obligatory Defensive Writing:

(1) Typical industrial servo motors implement sophisticated control over the velocity and acceleration profiles of the point to point moves. Anyone interested can look up the DS402 standard and take a look at the motion profile modes of operation.

(2) The PID loops implemented in servo drives are way more sophisticated than the canonical PID control loop equation (https://en.wikipedia.org/wiki/PID_controller). Nonetheless anyone who finds themselves manually tuning the servo loops for an industrial servo motor will surely find themselves setting proportional and integral gains.

Source: Commissioning servo motor drives is a part of my job.