I love the camera angle at 4 seconds, looking down from above. This gives a better perspective of how steep the run actually is. There's nothing like challenging conditions (e.g. steep, narrow, heavy snow) to stress our technique and bring out "old faithful" movements. So before we make critiques based on ideals, it helps to consider perspective. Dave is doing the task - just ski it. The second clip shows also shows good turn mechanics except that it's for defensive, survival skiing. There's a lot be said in favor of not getting hurt. "Normal" people do not ski such runs.
But we do. And no matter how good we do, we know we can do better. Many of us don't ski this kind of terrain often enough to intuitively know what tactics to use, not to mention be able to instantly adapt to unforeseen developments with the best tactic. This run starts wide, narrows and then widens again. It's interesting to note that the skiers in the second clip used narrow turns above the throat to make it easy to get through, then widened their turns after they exited the narrowest part. You chose to use the width of the available terrain, to let yourself use more turn shape for controlling speed than they did. When we are using shape, there are still 3 factors we are trading off: down the fall line speed variation, skidding and how much the turn finishes out of the fall line. It's possible to carve this run, but speed would increase. It's possible to skid and reduce speed and at any point on this run. It looks like it's not possible for most skis to carve a turn in the narrowest section and control speed. I go through all this BS in an attempt to imagine intent at specific points in this run.
So let's take a look at some specifics:

This is the second turn entering the chute. Angles lined up - tips, hips and shoulders. Inside hand ahead. Balanced against the outside ski. Approaching the first wall and thinking "uh oh - a little too fast". It looks like the inside ski is on a lower edge angle. This could be an illusion, but we're definitely going to be seeing 2 footed skiing coming up.
Feedback: More skidding at this point in the turn would avoid the coming collapse. Alternatively, finishing the turn more up the hill would give you more space for the coming left turn. For this option, you'd want to be driving higher edge angles here. The point here is tactics. If you'd started where you could see this section of the chute or had skied it a few times, you could have planned your line. Otherwise the trick is to be looking more than one turn ahead form the previous turn vs just reacting from this point.

Yikes! Throw on the brakes! Too much. See that bent over at the waist/rounded back? This happens in the bumps a lot. If you try to slow down more than the legs can absorb (uphill knee can't flex much more, downhill leg has to brace against the snow and can't flex or you'll lose stopping power), you must fold at the waist. Side note, look at the uphill hand below the hips and how much weight is back. This position is the beginning of the end of an ACL for many a lesser experienced skier.
Feedback: Keep the hands up!

There's a reason for the pedal hop turn. We don't really need a pedal hop turn here, but starting from such a folded over position, there's no way that inside ski is going onto the new edge above the fall line. And there's no time to waste because Dave is staring at that rock wall. So lift up that inside ski and ride that the uphill ski is relatively flat to the snow and behind the new inside ski.
Feedback: It's really hard to tip the skis into early edge engagement above the fall line on steep. It's ok to lift the ski when you're in a hurry and I love that you're lifting the tail. Pull the inside foot back and tip it more for increased turning power.

You're still not facing the inside of the new turn, but you do have your weight to the inside and now you know you're going to make it. Skis close together.
Feedback: You're screwed at this point. Because you have not started this turn from a countered position, the only pretty way to get to the inside of the new turn is to lean laterally into it. This just isn't going to work fast enough. So turning the lower body must be done. At least you're using femur rotation from this point on.

Phew! That was close. This is a good pic of turning into a countered position. Note that tip lead does not match countered hips and shoulders and no long leg/short leg. You're not balancing against the outside ski here. Before and after this frame the outside ski position is sloppy.And it does not get much better....
Feedback: Drive that old inside hand forward and guide with the old inside ski more. Stand more on the outside ski. Finish the turn more!

Ick! Remember when I talked about two footed skiing? There are a couple of very good reasons for why you had to do this. One is a very ugly rock wall dead ahead. Two is that outside ski wasn't doing much anyway. It was wobbling like crazy there.
Feedback: If you have to throw on the brakes here, do it with both feet.

Now we're talking! What a powerful position! The cool part about this pic is that you can see that there is only one more quick turn to take before the chute opens up. Now you've got more options and lot less danger. One option you could do from this position is to let the hips and upper body travel in a straight line to where they are facing, while the legs cross underneath. You'd come out of the next turn carrying way too much speed if the chute stayed the same width, but it's not a problem because you'd come back across below this rock outcropping on the left.
Feedback: I'd like to see that right hand leading a touch more here and held a little higher.

But wait, what is this? You're starting the turn in the middle of the chute. That's good. The skis are crossing underneath. That's good. You've maintained counter. That's kind of good (we'd rather see counter develop from the last pic to this one). But now you're going to choose to bleed some speed.
Feedback: Flex to release. Come across vs up and over.

There's that tiny pedal hop again. Almost a stem christie. Hey it works.
Feedback: It's all about tactics here. My bet is that mentally you were just thinking "whoa way too fast an exit out of the last turn" and "this is the last tough turn". I would have preferred to see :
option a) more skidding through the bottom of the previous turn,
option b) or have the previous turn finish more uphill,
option c) roll both skis into a shallower carve, going further down the hill and then ripping across into the wider section to slow down
option d) doing shallow railroad tracks from here down top the group with increasing speed.

The only problem is that you've come out of this turn without steering into counter. No problem, you can just park and ride through the wider section to control speed now.
Feedback: This should be just like a lane change drill.
Bottom line
There's no time to think through any of this stuff while you're doing it. It's got to be automatic and that only comes with practice and some level of comfort/confidence. All of the feedback I've given is easier said than done. David, you made a tough run look a lot easier than the other guys did. I know you have the skills to crank it up two more notches from this level. My bet is mileage alone will do a ton. Practice skidding more at the tops of the turns and finishing the turns more. Adding these tactics should upgrade your bag of tricks for these runs.