Thanks! I think for "canonical" lifts (squat, deadlift, row, to some extent military press) the vertical bar path is mathematically optimal, and for all kinds of lateral or sagittal movements you do more work with weak stabilizing muscles and load joints laterally too. Is it productive work that strengthens your core? Possibly, but it's hard to quantify. It it something that can lead to injury? Absolutely yes.

For more complicated lifts like bench press (J-shaped) or snatch (S-shaped), for example, I would rather set a "golden sample" path with a coach and compare to that.

It's unlikely to be the sole metric, especially given the inverse kinematics of different body types (long/short femur, etc), but together with bar speed, over time, it can provide a lot of good feedback.

It is not "absolutely" something that can lead to injury. Injury itself is difficult to define, and often the reason one experiences pain sensation is multifactorial. Within lifting contexts, generally the factor which has the strongest evidence for injury prediction is how sharply an athlete increases intensity compared to what they have previously adapted to.

No offense, but this post does come across as you only having a surface level understanding of the field. Especially surrounding injury/pain perception, I would be more careful of what you assume is true, there's far more nuance.