OK, so in summary:
- The ICs which were all "
supposed to be working" weren't necessarily working
- You were still trying to persevere with the IC sockets, and these were continuing to introduce intermittent problems
- There were multiple bad solder joints
- The oxide wasn't cleaned from IC leads before soldering or trying to fit them into sockets
I don't think sockets are any use with these old IC's and less than perfect finish on the legs. I also know next time to apply new solder to the legs and remove with solder wick to refresh the coating before soldering to the PCB.
Next you'll be telling me you used 100% tin solder instead of 60/40 tin/lead with a decent flux

If anyone cares, I'd recommend one of the
Kester leaded solders with 3.3% 245 no-clean flux, which is a great all-rounder for this type of work.
If you're trying to solder to heavily oxidised IC leads, I'm not surprised there are so many soldering issues. You can do the solder wick thing, or a much faster/easier way is a couple of wipes with a fibreglass scratch brush to expose a clean surface, then brush IC leads and PCB solder pads with a no-clean liquid flux pen immediately before soldering.
Sounds as though you may still have intermittent solder joints somewhere based on some of the occasional problems you're seeing. With the board running normally, you should be able to gently flex/twist it and it should continue to run normally. If it freezes, resets, crashes when you flex the board, then you've likely got other solder joint issues to track down.
Regarding the system not booting with the RTC/U178 fitted, than can happen if U178 doesn't have 5V at the supply pin (18). The system will run normally without the 32.768kHz crystal fitted, only the stored time won't increment, it'll stay 'frozen'. While powered on, the system RTC will increment normally since that uses one of the CIA hardware timers, but it'll always load the frozen time from U178 at every boot.