When you actually get the engine running, the fuel pump is able to function better than at cranking speed. So whatever is going on, I'd say it is right on the margin of fuel delivery.
I think if you took it into a shop they would put a fuel pressure gauge on it first. In my case I don't have such a gauge, so I have to "wing it" by evaluating the fuel flow at a disconnected hose. At cranking speed fuel should fly outa there rather forcefully.
Also, it would be relatively cheap maintenance to just go ahead and buy some fuel hose to replace all the hose on the system. If its the problem then you will fix it - if its not the problem then at least you've done some fairly inexpensive preventive maintenance.
Just a suggestion, I certainly don't know what's going on.......
1982 300D at 351K miles
1984 300SD at 217K miles
1987 300D at 370K miles