HOLY VERTICAL SPEED BATMAN! 
Hope it was empty…
Yes it was empty and light, this is the public display routine. A few years ago they made a wonderful video with fly-by camera’s showing it from above!
Not the same one but gorgeous anyway:
http://www.seattletimes.com/video/5026279436001/boeing-demonstrates-new-787-9-dreamliner/
What’s the TWR?
Quite a bit below 1. Honestly that doesn’t matter with those wings that generate a ton of lift, I could elaborate a little on the how and why’s but TWR is never used in civil aviation.
Usually you never go vertical in civil aviation as well.
Does lift really do that much on such steep climb angles?
Alright but this aircraft is empty, has only a shimmer of the amount of fuel it can carry, and engines that never deliver full power in normal operations(because that extends engine life).
So imagine this aircraft loaded with people, cargo and fuel rotating at about 110kts and getting airborne at 120kts. Now this plane wants to fly in a significant way at that speed when you remove all the limiting factors.
Climb angle is something that is only really used for passenger comfort, it has no specific aerodynamic property, angle of attack is what is important, this is the angle between the mean chord and the incoming airflow. Hence why fighter aircraft have a AOA indicator. The AOA determines when you get into a stall or not btw, it’s a funny thing.
Lift is generated by a pressure differential between the bottom of the wing and the top of the wing, lift increasing devices like flaps change the mean chord of the wing, thus changing the AOA(one of my teachers told me that wings roughly like a 20 degree angle into the incoming airflow). This is why the nose drops a little when you extend the flaps, the AOA increases and the wing stabilizes again by changing the angle.
So, if you go fast enough with any aircraft, you can fly it at some crazy angles. Although there is a big no no, which is the supersonic shock wave on top of the wing, this comes into existence way before the mach 1 sound barrier, and pretty much destroys your lift vector, I might be wrong on this one, but I think that this is where the mach tuck comes in, a nose drop when the lift decreases as you enter the transonic region and elevator effectiveness drops(where the shockwave comes into existence).
I think I am meandering a little here, but a few factors determine performance; AOA, speed, and lift modifying components.
On wing shapes, a straight wing generates the most lift(see Fiesler Storch), and a very sharp wing is most capable of flying in the supersonic regime(Delta wings are the ‘ideal’ compromise in this regard).
Man I could go on for hours but I don’t have time for that 
Climb angle is still relevant though in that the steeper you climb, the less lift will contribute to remain airborne, and the more thrust will do so.
But it’s the angle of attack that’s changing and coming into a stall regime, hence climb angle is more of a attitude thing(angle relative to the gravitational vector). Hence it exists but isn’t used much at all in the technical side of things.
Climb angle has HUGE relevance to me. I mean I get what you are saying but as I see it, the secret sauce to multi-engine survival is the amount of power the remaining engine(s) develop above that which is required for level flight. The reason we plebs never consider it directly is because our governing authorities have done it for us by establishing minimum climb gradients for departure procedures at all public airports with IFR capability.
Back to the thread!
I am totally with you as that is how it was taught to me! There this fantastic video of a wing in a wind tunnel from the late 70’, i think from shell, that shows that effect wonderfully!
@smokinhole, interesting! Care to expand a little more?
Expand? Not really. Only that at certain airports I consider the climb angle more than all other perameters. And sometimes (BOG, MEX, SJO on a hot day or tailwind) its a critical consideration even with all engines running–much more so when considering the plan should one quit. That 787 had no such concerns 
Widebodys, btw, have a big advantage in making impressive long-lens performance videos. They are designed to carry two or even three times their empty weight in fuel. When all but a smidge of the that gas is absent, well, things get real
Indeed they are, the A380 is one of my favourite aircraft to see take-off. That thing, fully loaded for a intercontinental flight climbs like a bat out of hell!
I’ve been thinking about your comment and I suppose I would agree with you, since the climb angle is a pretty good representation for a normal flight regime. Still doesn’t quite feel right to my brain though, but I am more used to consider those factors in a wind tunnel environment 