Fixed the blue bike this morning.
Found the electrical issue PDQ:
Replaced the inoperative brake switch (Read: safety necessity). All those spare parts salvaged previously are already coming in handy. It's not much to look at, from the outside:
Pumped up the tires and took it for my typical 4km test ride. Goes decent, considering; better than I expected from the two little 10ah batteries. Noisy gear box, normal for these motors. About what you'd expect from an entry-level electric bike.
Opened up the battery box after, so I can check the battery's real-world amp-hour capacity. Looks like the previous owner got tired of replacing fuses:
And considering battery options (seeing as there's little chance the 10ah batteries can put out the 35-amp limit for long):