This video does a very poor job of explaining the quality process. Inspection happens at multiple points throughout assembly, and most problems are caught before end of line testing. There are metrics on this and it's common to shoot for 98%+ of defects to be caught in process.

There are only a few problems that would lead to a scrap job - for instance if the body of the car is damaged to the point that they would have to cut and weld new panels, that vehicle would be scrapped.

But even for scrapped vehicles, all of the components are removed to re-build that vehicle.

Basically anything besides serious body damage can be repaired. But you are correct that it is hard to fix problems. Most parts are added in one minute or less - including the engine assembly! But it takes much, much longer than one minute to remove an engine.

At least some of the rejects are crash tested.