There are operator's options where the plane may be powered without the battery being turned on. I don't know of any US operators who have that option.
There is also an option where the APU will continue running after the generator is on line and the battery switch subsequently turned off. Once again I know of no US operators with that one either.
As to your original question, the aircraft I'm familiar with may have the battery switched off after external power is applied and many of the systems retain normal operating chacteristics. Some still require the battery switch to be turned on as you can see in the schematics.
If you turn off the battery and then remove external power it is my experience that the plane will accept external power and come to life without turning the battery on first. However things may not look quite right as some relays may change position while power is off and I promise you that you'll have faults latched in many components like the DEU's FMC's, and PSEU. Some of the displays can be very odd to say the least. One interesting fault is that you may not have any indication of right system hydraulic pressure. It's simply blank. This is because the BPCU will have a fault latched and the #2 DEU won't be operating. Until you start doing some system tests per the FIM the hydraulic indication is the only symptom. No Display Fault or anything, just a blank pressure display.
I've found that if you turn off external power with the battery switch on, then turn off the battery, our planes will not accept external power again unless the battery switch is returned on first.
Of course to apply power to the ground service bus you shouldn't need the battery on. (That doesn't seem to be set in stone either)
The correct procedure for applying external power on our aircraft is battery switch on first then external power. That way the power relays will be correctly positioned prior to AC coming on line.
If you turn on the battery first then the plane wakes up normally.
As to a diode missing in the diagram, probably not, unless your company has specified one of the options I mentioned above.