The only time that kicking the leg out is a problem is when it causes some other issue such as a balance or posture issue or maybe a timing issue. Many times, young kids who kick the leg out in front of them will lean back at the same time and that would be an unwanted posture change. But I don't really see any such issue with this pitcher (though that would best be judged from the front or rear - not from the side). So I am in disagreement with the coach in the video on this issue.
The coach in the video seems so concerned with the leg kick creating wasted movement in a direction other than at the plate that I am surprised he said nothing about the drop that the pitcher takes during the early part of his stride. That's wasted movement and energy directed downward. If the pitcher put a slight bend in his legs he could reduce the drop and truly get (almost) all movement toward home plate.
There's more to the front foot "rolling" or pivoting than just landing on the heal. Landing on the heal itself is probably caused by a lack of momentum. But there also seems to be some other things going on. On the surface, there also seems to be posture and balance issues. At front foot plant, it appears the pitcher's head and possibly his upper back (can't tell for sure from side view) are arched back toward 3B. If he doesn't recover from this, it turns into a balance problem later in the delivery.
I also see the shoulders rotating early. In fact, in the 1st pitch in the video, both the hips and shoulders have rotated at foot plant. In the other 2 pitches, rotation occurs after front foot plant as the should but the shoulders still rotate with the hips instead of in sequence. Early rotation can definitely lead to elbow issues.
Looking closely from the perspective of trying to identify something that could be causing reoccuring elbow problems, I notice two other things in addition to the early rotation.
First, the pitcher's head starts off facing the 1B sideline a bit and then, right before foot plant, it turns to face home plate directly. In fact, in the 2nd pitch in the video, it looks like the pitcher's head momentarily turns away from home plate a bit before turning to face home plate (though it's kind of hard to tell since the pitcher's arms get in the way). The problem I am concerned about here is that it appears that turning the head to face home plate during the stride may be causing the front shoulder to open early. This is not uncommon and is why I teach my pitchers to keep their head facing home plate from the get-go. The posture issue I mentioned previously probably combines with the head turning to help open the front shoulder.
Second, the pitcher's glove takes a high, arching path ending in a downward trajectory almost as if it's being pulled down. I think there is a chance this could also be contributing to the front shoulder opening early. I also see some inconsistency in the position of the glove arm at release. An inconsistent glove can lead to timing problems as well a control problems. Since timing problems can manifest themselves as early rotation, here is another contributor. Ideally, the arms should reach an equal and opposite position at foot plant. Once in that position, the glove should turn over as the glove elbow drops and the shoulders rotate. Forcing the glove arm to spend the time it takes to mirror the throwing arm buys time for the shoulders to stay closed until the hips have rotated.
So, my suggestions are:
(1) Add a little more momentum by moving forward a bit sooner - like right before the peak of the knee lift. Focus on pushing the front hip sideways toward the target.
(2) Fix the posture issue by keeping the head upright. Also, avoid turning it to face home plate during the stride - it should already be there before the stride. (Or, learn how to turn the head without also turning the shoulders.) Note that fixing the posture might lower his arm slot. That's fine. Some will claim it will reduce the downslope on his pitches but this trade-off is worth it because the health concerns are more important.
(3) Make sure the arms get to an equal and opposite position at foot plant. Alter the glove arm - not the throwing arm. Alter the glove arm's path if necessary. (Equal and opposite means the angles at the elbows and wrists are the same for both arms though not necessarily bent in the same directions. The glove arm doesn't have to hang out in this position for any length of time. It just needs to be there for the moment that coincides with front foot plant.)
(4) Once the glove has been extended out front, keep it there. Stabilize it somewhere over the front foot and then bring the chest to the glove.