No help delaying the manual reset.
No such problem when shutting off DC, so this means it works OK then?
Does it work properly when shutting off/reconnecting DC? No., So it doesn't?
I'm lost.
The problem stays @ 3.58MHz. Well I think that lowering the xtal does no more than
extend the time for the UC to initialize. Nice for measuring though.
According to the datasheet the 707 gives a resetpulse of
140mS minimum ->280mS max. This should be enough to let the UC perform 500K-1Mcycles @3.58MHz.

Are you using external RAM which is not sufficiently grounded? Or any other bad
or missing connection. Maybe the EA-pin (external address) has poor gnd.
At some point you mention a sequence AC_PF - delay (1-3S) - POR -> latched to incorrect state.
Meaning some device is taking over the data/addressbus? Do you have diodes connected anti-parallel
with your relays? Maybe when powering down the relays generate voltages that corrupt the 373's?
Just wild guesses.