Check fuel pump and relay, although I don't think this is the problem.
Most likely you have a corroded connection on the 0.4 ohm ignition resistor on the ignition. This resistor is bypassed when the key is in the crank position, so it will run so long as you are cranking, but die as soon as you release the key because the ignition is cut off.
I don't remember if it is the top or bottom one on the w116, but check both for condition and connections. Corroded electrical connections cause all sorts of strange things.
1972 220D ?? miles
1988 300E 200,012
1987 300D Turbo killed 9/25/07, 275,000 miles
1985 Volvo 740 GLE Turobodiesel 218,000
1972 280 SE 4.5 165, 000 - It runs!