Do I understand how this Digital Integrator circuit is supposed to work?
I must confess. 

I did not check your circuit in detail, but I think you understand the concept OK.

A counter is a Digital integrator because it accumulates the events per unit time, if operating at a fixed frequency.

We can enable and disable integration by the clock control and we can vary the gain of the integral by changing the speed of the clock or simply just count events. 

We could also change the gain, as in this case, using a parallel adder and accumulator to integrate or count the total number of events over time.

Then you can get fancy and use an ADC to create the values to control gain of the integrator or accumulate the events per unit time or some period of time.

You can integrate in a spreadsheet too or just with an Op Amp and capacitor feedback and resistor compared to a Voltage reference or use a microwave integrator.  

There are many ways to perform this "integration of an input over time" , each critically depending on the specifications and tolerances needed.