Well I had a bit more than five minutes today. I did manage to get an LED to light but I needed to rectify the rx with Schottky diodes: using an LED to self-rectify didn't work, not sure why.
There was little if any benefit using a full wave rectifier compare to using a clamp diode plus single rectifier diode.
I could get it to light visibly with about 1mA at a distance of 2cm, so as you say, it's still near field. As you need to quadruple the radiated power to double the distance, I'd need 16 to manage 8cm(!) however I could also add gain to the receive element too in a similar fashion.
It looks like there are 900 transducers in the beam forming unit. Say it's 1024 to make the maths easier. That gives a possible gain of 30dB over a single transducer by my calculations and a range of 64cm.
20Vp-p is the maximum the transducers I have are spec'd for. They don't look exactly like the ones in the video because there is a plastic protective mesh in front of mine but the size looks similar. At 20Vp-p I am measuring (real) power into the device of around 100mW. At 2cm distance, I had about 2mW, giving it a 2% efficiency. However, at 2cm this subtends only a small solid angle from an unmodified beam tx, so perhaps only 15-20% of the transmitted power appears at the rx anyway. So beam forming and reasoanably sized apertures on the receiver are essential facets for this to work.
It'd be an interesting experiment to put a larger array together for both tx and rx on a parabolic surface and see what can really be achieved, and at what distance. I only have 30 transducers in stock though!
I was thinking about their brick phone attachment, and assuming it's real, in itself that would have to have a reasonably large aperture, so perhaps 40 sensors. Each would need MPPT and power aggregation which although not rocket science is going to be expensive to fabricate.
That camera thing is an Nvidia Jetson which looks like it's for visual device tracking. They do like making hard work of things! If it needs visual indication of where the target device is, and the sensors are on the rear of the phone, the phone will have to be used face down for a ceiling arrangement, and you won't be able to hold it in a normal fashion to make a call or use the screen. Even wall mounted, assuming nothing's in the way, you'll have to figure out new ways to hold your phone.
In its current form and key use, as a phone charger, this remains practically speaking a non-starter.