I've seen bad DMFs that would rattle on start up for quite a while. A quick blip of the throttle will usually make it stop.
You have mentioned that the last owner replaced the pump and injectors. You didn't mentioned whether or not the timing was ever set correctly or even checked. That certainly wouldn't hurt. His timing was definitely off due to the blue looking smoke coming out of the exhaust. Was that the source of his rattle and rough running? Maybe, maybe not. That sounded just like a bad DMF to me, but it's hard to be sure over a video. He does have a ZF5 transmission. He was also wrong when he mentioned that air bubbles coming from #1 injector return to the filter means that the injectors are leaking air. That can't happen because they have fuel running through them, not air. They can't suck air because they're under pressure when the engine's running, not suction. Now if there was an o-ring leaking, letting the fuel drain back to the tank when the engine's shut off, then you could get some air going through that line when the engine is first started. If that's the only leak, the bubbles would clear up after the engine runs a while and pushes the air out of those lines.
The DMF rattles because it has springs that act as cushions between the two pieces. When it's in good shape, the springs will compress and release while keeping the two pieces working together like one piece (basically). When the flywheel gets to be "bad", it's almost always the springs get weak and then they stay compressed which allows the flywheel parts to slam back and forth, causing the rattling noise. The two pieces that are slamming back and forth are also throwing the engine out of balance for a couple of reasons. 1) the balanced parts are no longer in the proper place to balance the engine. 2) you have a heavy piece of metal that is moving against the rotation of the engine and then quickly moving with it faster than the engine internals are moving. That's why the rattle mostly happens at idle speeds and slightly above.
One more thing to point out about that video. Notice that he said that he does not have any air bubbles going through the clear hoses. That completely throws air out of the equation for his example. If your rattle sounds exactly like his does, that just adds one more point in favor of a bad DMF in my opinion.
If you want to eliminate the inspection cover as a source of the noise, don't add more screws to it, just remove it. Then it can't make any noise. You can also see if it was being rubbed by the flywheel because it will have clean scrape marks from being rubbed. While yo u have that cover off, you can check your flywheel. You can grab the clutch and see if you can move it back and forth in a circular motion that runs along with the flywheel. If you can move the clutch, the flywheel is bad. I hope that makes sense.