You can start by scoping GPIB data lines.
If there is life you can expect some boot info.
Next you can scope all possible chip select signals, maybe you are lucky and only few appears.
Skip memories, they are tested or not and passed enough.
All parts don't need any actual initializing, but they still must, or should be found.
If you want to verify what is really happening you need to figure out what is coming and going.
For that one possibility is a logic analyzer, but it is much work for little gain.
And you still need to check all programming manuals to find out what all those things mean, and then find out that the program is actually doing exactly what it is suppose to do.
The block diagram indicates that there's only one pair of bus drivers, so data part is fine since CRT has meaningful text, that must go through the data part of the program bus.
All other parts are unknown, but can you decipher anything from the text on screen.
Maybe you should start from the assumed end.
One can assume that checking ROMs is so easy that it is done and failure would be indicated differently.
But if it happens to be so that lower part of the block diagram parts are getting attention does it mean that upper part is already done and ok.
TI DSP is a normal part so that is easy, but Advantest bus controller is trickier.
1st left memory block of the lower part of the block diagram, is it MEMCRT?
If so then check their _OE(16) pins, they are 8x memory chips left from bus connectors.
If _OE doesn't go down those memories are not used and text on screen is directly written to CRT controller.
If so then it's possible that the text on screen is not present in ROM in normal ASCII text form.
But this or that reading ROMs out is always beneficial.
In case you don't have a gear for that near by, don't use too much effort, it doesn't help you in near future.
In case you have hard times accessing pins when the machine is in one piece check those IC pin minigrabbers.
Some scopes have multiple triggers, if you want to figure out the initializing order of chips.
Simple dual D-flop circuit will do it also, there CS signal flips the other flop's set or reset, adding few LEDs is not a bad idea.
Then you just flip pairs to find out what is always later.
Chip select signals are multiplexed.
So following the trace back you should find from where it is coming, it is not a multi use signal, if not for multiple equal chips, like memory chips.
The problem can be also that multiplexer.
Checking (de)multiplexers and multi line transceivers can be tricky.
For that you may need a logic analyzer, and 8 bits is not very wide.
I'd probably skip MCU stuff, few latches and D-flops with LEDs are much more versatile for single purpose.
But there are so few peripheral program bus chips that you'd probably be fine with cheap 8 bit USB thingy.