Since the amount of time dedicated per process is variable, it would seem that programs should run at non-consistent "rates".
They do. In modern Operating Systems each program/process is assigned a slice of the available CPU time. But most programs, especially services/daemons, do require less time than assigned, so other programs/processes can utilize more.
Yet many programs in which this would be an issue, such as games, video players and drivers seem to run perfectly consistent.
You're right: it seems that way.
But every serious player does know about the deteriorating effect of background processes/programs. The general rule is: the less background processes, the more CPU speed/capacity is available for the (foreground) game. So usually, in a "game" situation, most of the CPU processing time slices are not used by other processes; and the main consumer is the game.
The only way to avert this is to tell processes how much of some unit of time is spent executing its code
Here you are mistaken. The concept of Preemptive Multitasking avoids the necessity of cooperation of the processes with each other. What you assume is happening is called Cooperative multitasking which "is rarely used in modern larger systems".
Both are different approaches, here is a relevant part of the definition of Preemptive Multitasking:
In computing, preemption is the act of temporarily interrupting a task being carried out by a computer system, without requiring its cooperation, and with the intention of resuming the task at a later time.
This is the concept that you erroneously assumed to be invalid:
[...] since I'd also imagine querying the system clock periodically would be very inefficient, and would have to happen very frequently.
Preemptive Multitasking is exactly that:
interrupting every process at a given time induced by the timer interrupt and transferring the execution to the Scheduler (which then selects the next process to be executed).
Preemptive Multitasking is used by most modern Operating Systems like Windows, Linux and MacOS.
In-Depth information about multitasking approaches can be found here at OSDev.