1995 Volvo 850 GLT long crank

[misha]

...It all looks good, except:

When engaging the starter, the fuel injectors won’t fire, but they get a small burst of activity just as you are turning the ignition off.

Fuel injectors are getting power from gray relay(at cooling fan carrier).It's main fuel injection relay.It feeds a lots of sensors with key in position II along with injectors and coil.

You also mentioned that shop determined that there is also no spark.Grey relay powers the coil too.

The fact that injectors become active just as you turning the ignition OFF,indicates that ignition switch might be faulty(mechanical or electrical part of it).

[abscate]

I missed this

They spent several hours diagnosing and found: no power to the injectors when cranking.

Fuel pressure is good, but no spark. They found they can easily push start the car and it reliably starts immediately. They went through every sensor, all grounds, the ignition switch, checked all of my components. Put the sensors on a scope when running.

From here, a competent shop would take 10 minutes to diagnose.

No spark and no fuel points to a B+ failure to these and a circuit trace strategy back through the Main relays and ignition switch

The fact that injectors become active just as you turning the ignition OFF,indicates that ignition switch might be faulty(mechanical or electrical part of it).

..and misha is on point as always

[jimmy57]

The fact it starts instantly pushed puts it squarely in the dirty trans and bell housing face or faulty starter territory. Starter current draw has been ruled out.
The barely adequate suppression fails when magnetic flux goes up or the suppression goes down. When the key is released and starter current stops the magnetic flux lines go away and the momentum of engine creates enough signal for that one injector cycle.
To clean the contact surfaces you will loosen every engine to trans bolt all the way around 5-6 or maybe a little more turns and then use a rifle cleaner brush and brake cleaner spray and scrub the lower half. Now put a jack under oil pan (on a piece of 2x6 or metal plate--spread the load so as to not punch a hole in pan) towards trans end and jack it up until the gap closes down at bottom which will mean the top side is opened up and then do the same cleaning. You may have to lay on top of engine to do the rear. You may have to pry the engine and trans apart initially as the dowels may stick the two together. Now tighten all bolts and try it.
The coil wire and crank sensor wires could cause this also but I never saw that happen myself. Not any of the ones I ever was called on had those two routed in a way to cause the problem.
When the cars were newer with low miles it was always after the engine and trans were separated for a rear main seal, clutch, or trans repair. In the last few years aging starters brought up starter high current as a cause. The ones with starter issues may not crank over slowly enough to be noticeable or to be noisy.
Some with these issues will on occasion set cam position sensor codes. ????

It is not absolute that normal current draw (<150A) rules out starter but I would do the surface cleaning before I bought a non-returnable test starter.
Oh, ground leads tightened or even added would seem to make sense as a fix but no such luck. I have even known of foil wrapping and other suppression tricks to not work but cleaning engine and trans surfaces BINGO!