Maybe a possible solution 

Чтоб получить тот синус, нужно очень точно настроить на 20 килоом потенциометре и с 4 килоом потенциометром может легче будет, это в самом начале он получаеться и тогда катушка начинает жжужать. При ниже напряжении он вроде бы легче получаеться. Но может и не в нём тут суть, а может просто Вадик наткнулся на какую-то ошибку микросхемы и поэтому получил самозапит, но я повозился на своём сердечнике стержне и самозапита как небыло так и нет, потребление около 100-150 милиампер. При питании 3 вольт потребление может даже и 200 милиампер или выше и никакие подстройки никак не спускают его. Поэтому может тут суть в медных лентах, я их не мотал.

So the bottom winding is split into two parts and the top winding is not.
So that makes 3 coils and the text states that in the 1^st sentence: "три катушки" = 3 coils.

Is this Russian text translated into English somewhere?

hi MenoFather,

Have you try to swap only one of the winding be it primary or secondary side to solve the issue and measure current consumption again during tuning.
It can be a common mistake normally made on self-made transformer with 4 wires(2 pri, 2 secondary).If connected wrongly the circuit will never oscillate.


By observing video 4 you can almost hear the  ticking noise base on slower frequency signal produced by I/C:4069.Do use this frequency of ticking sound as a guide for your circuit.

Have you look into the counter poise beneath the Ferrite core on the copper track which looks near to a square shape(-).

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I can't confirm if this do plays a part-----------------------
I observe the copper\brass screw thread is soldered onto large copper track(square shaped).

Do observe the screw at the center of pot core  is metallic instead of plastic.The top half is nicely insulated using the "plastic insulator" with the metallic screw going through it which is also similar shaped for TO-247 heat-sink insulator\mounting kit

That's a nice hand drawing!  I could not draw sth like that without a computer.
It is still not perfect because it is not clear which junction connects with the foil and which connects with the wire.

If only Vadik Guk could draw like that !