As suggested in the video from the last log, I finished soldering together one of the bad boards. Today, I took a bit of @K.C. Lee's advice and tried standing the shift registers up on one edge:

I enjoyed avoiding six point-to-point solder connections. Securing the chip first was a big plus, too--the previous method required a few extra steps and was difficult to get right on the first (or second...) try. The four side connections were, once again, the most difficult part of the operation.

After some de-fluxing, this is what the two boards look like side-by-side:

I also temporarily soldered wires onto the six programming pads of the first board and tapped the board a few times to re-check the LEDs (the newer board was checked in a USB hub):

So why the '-ish' in the post title? Well... I haven't actually tested the functionality of the boards. As in, I haven't sent the shift register a specific series of pulses and observed the correct result.

The second '(-ish)' in the title is because this hardware iteration is destined for the wastebasket: it requires the user to install 67% of the ICs upside-down. Not a good design! I did update the schematic and board layout but haven't double-checked the work yet; I want to be sure I include some of the other complaints I'd recognized previously. And I want to make sure the circuit works, so that means the next step is [finally] working on firmware using the two existing prototypes.