As for the power rating: Mini-Ringkernrechner, a popular software in the German ham radio community, says that even the smaller T50-2 core would easily handle 10 W with a 100% duty cycle. The flux at 23 Vrms (10W) will be 2.3 mT, while the core can handle 5.7. And that is for the T50. My guess is that they are using T 68 cores because otherwise you will have to use 0.4mm magnet wire, which might increase losses. And actually the max flux will be lower.
I am currently building a variant of this antenna design with T50-2 toroids and 0.4mm wire. The losses due to the smaller wire will likely not be very relevant in comparison to the ground losses.
Heinz, HB9BCB pointed me to the excellent works by Gerd Janzen (DF6SJ) on short antennas, and Wolfgang Beer (DK2FQ), namely this one (in German):
By means of simulations, he shows that elevated, resonant radials are much more efficient that those flat on the ground. So I guess that 2 elevated (maybe sloped) radials will improve the vertical remarkably.
73 de Martin, DK3IT