I also am confused. I have had a sleepless night thinking about this.
If the wire is 20 m long then the following is the (very) approximate end impedance.
RF Wavelength Wire length in waves End feed impedance Approximate Z
80m 1/4 wavelength Low 50Ohms
40m 1/2 wavelength High 2700Ohms
20m 1 wavelength Low 200Ohms
15m 1 1/2 wavelengths Med 700Ohms reactive
10m 2 wavelengths Low 100Ohms
So what do we get if this wire is fed with a good match?
On 80m we have a low height quarter wave, we should expect high angle radiation but some dB loss compared to a dipole because of the unbalance.
On 40m we have an end fed half wave, the feed impedance is high and because of the low height we should expect high angle radiation.
On 20m we have a full wave aerial, the vertical radiation is suppressed and end fire is supported, we have gain towards the horizon.
On 15m we do not have a clear picture, the impedance is moderately high and reactive. again the vertical radiation is suppressed and end fire is supported, we have gain on the horizon.
On 10m we have 2 wavelengths and a lowish feed impedance the vertical radiation is suppressed and end fire is supported, we have gain on the horizon.
So provided we can align the wire, we can enhance the DX radiation on 20, 15 and 10 m.
Then we come to the Transmission Line transformer.
It is difficult to see how a fixed transmission line transformer can convert these very different feed impedance to a low SWR for the rig to feed it across 80 - 10m.
Looking at the circuit I have some questions:
I do not see the purpose of the capacitor, is this to tune the primary of the transformer to resonance on one band?
In my experience transmission line transformers rely on tight windings of bi-filar, tri-filar or higher order to maximize the inter-coil capacitance and enough turns to achieve inductive coupling at the lowest frequency in use. The core only acting as a magnetic coupler.
So looking at the photograph it looks as though we have a 2 turn primary loosely coupled to a possible 2/3 turn secondary, connected to a 13 turn inductor. The crossover connection is also unexpected. This device is often used on choke baluns to reduce the input/output leakage capacitance. I cannot understand its purpose here. At 10 W the peak voltage on the end of the wire is about 250V peak, does the insulation on the torroid needs some consideration?
All that said, the 20m long EFHW for 14MHz does intrigue me. I usually use inverted vee link dipoles that are resonant on each band of choice. Having an end fire wire on 20m would be a worthwhile addition to the rucksack. I just need to get my head around feeding the wire. It is certainly much simpler and probably out performs an elevated Ground Plane in the chosen direction.