Hello,
Interesting question. I have quite a lot of experience repairing military equipment. Usually it is high-tech, complicated, no schematics and zero information on the internet. Some repairs take me months, literally.
So your question is what to do if you are faced with unknown. Well, I guess there is no magic. I don't know how these companies work which fix bare boards. I know just what I do.
First I try to gather at least some information about equipment. I talk with people, who use it. I look for information about similar equipment, scientific papers to familiarise myself with theoretical side if it is something new to me. So I read a lot.
I get back to the thing and look basic things, which are likely to fail, likely to experience stress, electrical, thermal or mechanic. Like for example power mosfets or fuses, input, output sections. If quick inspection with multimeter does not show nothing obvious then I start looking and reverse engineering by divide and conquer method. First rough, then going into more details as needed. What are inputs, outputs, power rails. After I have power rails and voltages confirmed, I can power it up. Look with thermal camera, measure voltages. If nothing is again obvious I go with more detailed reverse engineering.
I start drawing structural diagrams and collecting datasheets of all unknown components, IC's. If needed, I build some circuits to simulate inputs, write some test software. Also capturing internal board comunications, like I2C, SPI or whatever there is. Even remove and test separate components. This can really go deep and long.
This is all generalizations... You never know what will happen.
If I need to vote for the most useful tool it would be diode mode on a multimeter (it should go to at least 3V). For example, if there are many inputs like a data bus, I can go with diode mode (sometimes both polarities) to compare them. Compare other things. If there is a connector pin and I don't know where it goes, but it likely goes to an IC, I put my diode mode on that pin, get a voltage drop, go quickly with hot air over IC's and if I hit correct IC, voltage drop lowers, so it helps tracing. Also in some cases diode mode can be used as a current injector to simulate a signal. And more
Another useful thing on a multimer is LoZ mode. For example, there is 3.3V on a pin. Is it output or input with pull-up? Switch quickly to LoZ and if voltage drops a little, it is likely an input. And if pin is 0V? Is it output or pull-down? You guessed it. Diode mode injection. If 1mA can rise it, it is likely input with pull-down.
It is endless topic really. But if you like puzzle solving and don't get frustrated easily just try. Likely it will not be a one day repair, but if customer is giving you some time, go with courage and explore.
But if it is possible, always ask for the whole thing. That saves
a lot of time.