Edit: this continues on with the counter level, where you can use the register but it does not behave like the typical set of edge-triggered flip-flops (or transparent latches) that one would normally use. My best attempt at coming up with an explanation is that it's like two-phase edge-triggered logic.
Yeah. The flip flop, and its corresponding register version would be a DFF with enable pin. That is not actually too abnormal. The traditional implementation is taking a normal DFF and adding the enable feature either by gating the clock or by adding a mux in the input path, that uses the enable signal to select between the input or the existing output.
Everything started making sense after I thought of "store" as "enable", and I made it to the end pretty quickly after that --- except for the "bonus" multiplier level, which is definitely possible but seems like it would take about 10x as much space as the canvas allows.
Edit: this continues on with the counter level, where you can use the register but it does not behave like the typical set of edge-triggered flip-flops (or transparent latches) that one would normally use. My best attempt at coming up with an explanation is that it's like two-phase edge-triggered logic.