The ICs were surface-mount flat packs onto boards in the modules. They were welded, not soldered. The modules plugged into the backplane connectors, which were wire-wrapped. The modules and backplane were encapsulated in plastic (epoxy or polyurethane) for flight. (This AGC was for ground testing, so it was not encapsulated.)
The non-logic circuitry was built with cordwood construction, i.e. the components were inserted into holes in the module. Point-to-point wires were welded onto the components on either side. These were also encapsulated in plastic.
I don't know if DIP integrated circuits would have been more reliable, but the DIP wasn't invented until 1964, too late for the AGC.
The Saturn V rocket contained a Launch Vehicle Digital Computer (LVDC), which was completely different and built by IBM. It also used surface-mount modules, but IBM's hybrid modules, not integrated circuits. So the AGC wasn't the only surface mount computer at the time. (Does anyone have a good history of surface mount?)
> if there were anyone who could do it, it's him
Honestly, Mike Stewart is the AGC super-expert who made the restoration possible. He knows way, way more about the AGC than I do.