I am no big expert, but I try to watch lifts from the perspective of fixing big, common technique errors, and then narrowing things down from there.
For example, very new lifters will almost always have one or more of the following:
- arm pull
- loose or rounded back
- hips shooting up
- bar crashing into knees (not "pushing knees back")
- all kind of second pull deficiencies, like not using the hips properly, etc
So when I am watching a newer lifter, I watch out for that stuff. If that stuff isn't popping up, then I try to look at their general positions at each point of the pull--where are they positioned at the start of the pull, at the knees, at the end of the second pull. Checking their feet to see where they're shifting their weight is helpful as well. For analysis purposes I haven't found looking at bar path to be helpful in real-time, exception being to confirm if the bar is swinging out during the snatch. Watching the bar doesn't tell me what they're doing wrong with their body that's making the bar move in the wrong path.
The third pull is often the last thing I look at, because in new lifters most issues with the third pull can be fixed by fixing the first and second. Exception is timing issues, like the clean crashing, but those are very easy to diagnose. Generally any lifter who is pulling the bar a mile high but getting squashed and rounded at the bottom of the clean is having crashing problems.