Sounds like you really have isolated it to a fuel problem.

Both relay and pump failures are common.

Both can be intermittent.

While it is not starting, try simply whacking the fuel pump with something hard but blunt (handle end of big screwdriver, for example). If it starts, you know the pump is the problem. This sounds silly, but I have 'nursed' several cars with sick pumps home in this way, including my 300E once.

If that does not work:
Have an assistant turn the car to the On position while you grasp the pump. You should feel (and hear) it come on for a couple of seconds. I believe the key needs to go all the way back to lock each time to repeat this test.

If it is not coming on, check the voltage at the pump during this (2-3 second) window. My digital voltmeter will not establish a reading that quickly. This is one time where an el cheapo 'lightbulb" type voltage detector is better. If it is getting juice but not running, the the pump has seized up.

If it is not getting voltage, odds are the relay is bad.

Good luck.

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1986 300E 5-Speed 240k mi.