below is a photo of Abakan a son of Absent and an Anglo-Teke mare. Abakan was trained and showed by Yelena Petushkova who was herself claiming that Filatov's training philosophy was to break the horses by hours of training using spurs and sharp bits to force the horse to do what he wanted.

Filatov himself was a cavalry officer under Marshall Budenny who from 1920 was managing horse breeding in the USSR. Budenny assigned a horse called Ingas to Filatov and encouraged him to study dressage in order to show Ingas. Filatov placed 11th on Ingas in Stockholm Olympics in 1956 (best of the Soviet riders).
I think though that the ride Filatov and Absent did in Rome is one of the most brilliant rides, still, made in Olympic dressage, that is why Absent is still a known name among dressage riders today.
To me it seems impossible that methods and a riding style that Petushkova mentioned was used to acheive what Filatov and Absent did. I think that it was a combination of an outstanding horse, a brilliant rider and Teke magic.
Filatov himself said that the dressage horse should be graceful, beautiful, a piece of art and that you should pick only the most elegant horses, like Tekes for dressage. And today he is proven to be right as the dressage horses are becoming less bulky and more elegant.
He also claimed the Tekes can be stubborn to begin with to train but once you overcome the stubbornness by training they become mind readers and totally faithful to you. I think that this is what Filatov experienced with Absent, the mindreading ability and the total faithfulness and this also what the audience in Rome felt like shivers up and down their spine.