If your fuel system is similar to my W126...

So far as testing each component, that would be a LOT more work and expense - plus you would not know what specifications would be acceptable for each. Best to follow the Mercedes factory service manual testing procedures for the entire fuel system. That will pretty well test everything in the fuel system.

The thing is there are various tests which are buried in the service manual [at least for my car]. So may need to hunt around - fuel system, emissions, fuel system venting... I forget where they all are.

But there should be a fuel pressure and fuel volume test (how much fuel is flowing). That will test to see you have the proper pressure. And the volume test will show that the fuel system can flow enough to provide all the fuel required. For example with a clogged fuel filter or one pump not working (on two pump systems) you may get the pressure, but not much flow or volume.

Then a fuel "internal leak" test. That is connecting a pressure gauge to the fuel distributor, starting and then shutting off the engine, then seeing if the fuel system holds pressure. If the pressure quickly drops, it could be "check valves" or the accumulator.

The accumulator is like a mini pressurized storage tank. It stores pressurized fuel and that is available for start-up and perhaps if you suddenly floor the accelerator. Also it can "dampen" noise.

The thing is... As fuel enters one end, the opposite end needs to vent air. And that air is vented to the fuel tank. If the accumulator is bad - has a leak (or internal leak if you will), then fuel will flow right through it back into the tank. That would show up in the internal leak test - the system would not hold pressure with the vehicle off. And may have lower pressure with the system running.

Also when you first turn on the ignition before starting the engine, it briefly turns on the pumps to "prime" the fuel system - you should see the fuel pressure rise.

And then there is an "evaporative system" - venting and so forth. Tests for that. Perhaps in the emissions section?

With the pressure gauge, you can perhaps rig it up temporarily so you can see it when driving. Then drive around and see if you are getting the correct pressure and if there is no pressure when you stop at a light and the car dies (as you suspect).

If it has no pressure at that point, I would rig up something to monitor the fuel pump voltage. See if it is losing electrical power. And maybe first check the fuel system electrical components. Clean the plug in pins on the electrical modules under the hood, etc. Also the overvoltage relay if you have one of those.

I found a fuel pressure gauge with all sorts of different size fittings at Harbor Freight for not too much $80 or $100. Be SURE you know what size fitting you need and that the testing kit you get has that size.