300SDL Vacuum for Engine Shut Off and Doors Lock Not Working
 

So as the title says, the engine will not shut off after I turn the key in the ignition switch.
However, as you can see in the video below, there is still a vacuum. Any idea what it could be?

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It happened after I replaced I removed this device that had a cracked elbow tube as you can see in the attachment below. The new elbow tube can not hold a vacuum. Could the faulty elbow tube be the problem? Or improper installation of the device? How can I do further diagnoses on this vacuum problem to pinpoint the issues and take care of them?

The device is the vacuum cut valve, which controls a vacuum signal to the transmission and has to do with the quality of the shifts (soft / hard).

The elbow connects a vent line, so that isn't your issue.

You probably disturbed some other vacuum line, probably one that connects to the fuel shut-off valve on the top of the injection pump. It is very common to break the T fitting on the main vacuum line (between vacuum pump and brake booster) and then there is no vacuum supplied to the vacuum switch on the ignition switch. You may also have pulled off the vacuum line at the shut-off valve itself. It is also common for that shut-off valve to fail. You can use your mouth to test the valve - sucking on the vacuum line directly to the valve should cause the shut-off lever to move from up to down.

Make sure the brown vacuum lines aren't broken. One comes off the vacuum booster line and goes into the dash to the ignition switch, the other comes back out into the engine bay and connects to the dome shaped vacuum pod on top of the IP above the STOP lever. At this age they get fairly brittle and break.

Your door locks will be unrelated to this problem. The door locks on these cars don't connect to the engine vacuum source. They are operated by a vacuum pump in the trunk. Make sure Fuse 15 is good. If you listen to the trunk when you operate one of the front door locks or the trunk lock, you should hear a "whirring" noise as the pump runs.