Unoptimized and low GPU load

Is CPU RAM in dual channel mode? You can verify with CPU-Z app.

Ryzen 9 5900X has two CCD dies. Try disabling one of the CCD dies and see if that helps.

I have a single channel 32 GB, but I don't think buying 16x2 will change anything.

I already tried disabling it and playing on 6 cores, but nothing changed 

I don't think its a hardware issue.
You can enable advanced performance telemetry in the game to make sure that the game actually recognises your GPU.
It seems to at least dedicate video memory to your GPU so it should find it but its never wrong to reinstall drivers with DDU. I had to do it during the beta on my Nvidia card. 
Do you have hardware acceleration on in windows?
There is reports about EA overlay causing issues as well so id suggest you look into that as well.

Had the same issue with my 5600x and DLSS, upgraded to 7800x3d and my 5070ti is unleashed

You're playing in single channel. Switching to dual channel will definitely improve performance.

But the strangest thing is that in warm-up (shooting range) I have 80% GPU load and 250+ FPS, but as soon as I switch to any online mode, my video card refuses to work at 100% and the entire game load goes to the CPU

Have you tried to apply the CPU fix in the user.cfg file? 

Yes, I have a file with such date :

Thread.ProcessorCount 12
Thread.MaxProcessorCount 24
Thread.MinFreeProcessorCount 2
Thread.JobThreadPriority 0
GstRender.Thread.MaxProcessorCount 24

Did you ever find the fix to this? I have R9 5900x with RTX 4060ti 128 GB ram dual channel, Same issue.