I accidentally found a solution to this problem. I did not immediately check my old screwdriver, then I already found the backlash of the cartridge. It was possible to drill with such a tool only at very low speeds. I ordered a new one powered by the mains, which turned out to have exactly the same defect. There are many videos on the Internet on this problem, but I did not find what I needed. I decided to figure it out myself, remove the cartridge and replace it with a domestic one - it would suddenly work.
We eliminate the beating of a screwdriver
To remove the cartridge, unscrew the bolt with the left thread. Unscrew the key clockwise.
The bolt had to be unscrewed with effort due to the applied thread lock. Even more problems were with the cartridge.
I had to clamp the screwdriver in a vise, insert the key at 16 and using the emphasis at high speeds, only with the third attempt did I manage to remove the cartridge, which was very tightened.
Domestic cartridge did not fit the thread. It remained to buy a new one. But first I decided to try again. I replaced the old cartridge without tightening it. Tightened the bolt and tried to put a drill. It turned out that the backlash was gone.
Then I began to understand why, after removing and replacing the cartridge, the play disappears. It turned out that there is a backlash until the cartridge is clamped. The defect may remain or disappear depending on how the chuck is pressed against the surface of the screwdriver. Metal surfaces at the installation site are machined on the machine and must be installed evenly. Backlash appears when very tightening. It turns out that part of the surface is pressed harder and part is weaker. There is a slight skew over the entire length of the cartridge and, as a result, the runout of the drill.
In the same way, I eliminated the runout of the cartridge in a new screwdriver. There, the bolt is screwed with a Phillips screwdriver. He removed the cartridge and put it in place, without clamping too much. Backlash in this case has disappeared.
I remembered that a neighbor complained about the curve of the cartridge, who bought a drill in the OBI store. We unscrewed the cartridge from the drill and put it in place, carefully locking it. The defect has disappeared. So this method can be useful for those who have a runout of a cartridge in a screwdriver or drill.