The IRFZ34 is a N-channel and the IRF9Z34 is an P-channel -- quite different little creatures.

Yeah, the "anonymous source" knows that a tube of IRFZ34's somehow ended up in the IRF9Z34 bin...
"He" thought he had accounted for all of them but was apparently wrong.

Shoot that "anonymous" source an email.. I... um... I mean 'they' will take care of this.

Plain 7400 series parts are getting harder and harder to stock and cost has gone wayyy up.

74F or 74S for high current loads.
74S has been phased out and replaced by 74F.... but are both identical (as far as we're concerned).

74LS or 74HCT for low current loads.
74LS is slowly being phased out in favor of 74HCT. Seems each week I get another DMSMS report listing more ICs hitting the obsolete status.

Solenoid drivers have a 560 ohm pullups on outputs. That amounts to about an 8mA load which maxes out the 74LS parts and exceeds the limits of 74HCT parts. Need to use high current 74F or 74S here.

Switch drivers have 4.7K ohm pullups on outputs. That amounts to about a 1mA load -- any of the above parts would work. I would advise the 74LS08 as he has a little bit more 'umph' than the 74HCT08.

"I thought I could put the system in lamp test mode, put my logic probe against the different outputs of the 7408's and see the steady on/off of the lamp test, but no dice. I see a steady high on the pin that takes in the blanking signal, a pulse on the pin that goes to the PIA, and a low on the output."

You're testing it correctly.
The high from the blanking (normal) anded with a pulse from the PIA (normal) and the 7408 having a steady state -- not normal. I'd replace the 7408.
The good thing is that you have pulsing from the PIA - he's not dead.