Sorry for late reply, didn't get the notification.
Printer's main board is from Anet A8 (it's modded Anet A8 ).
The head heats up, I can see it on the screen and measured temperature of the heating element.
As I previously said, I managed to get 1 succesful print after slicing in Cura. I printed few first layers of this model again and it turns out it had the same issues on first layer, but because it had big solid bottom layer, filament 'cached' the bead. This first layer was a bit deformed, but other layers looked good.
Adding T0 don't change anything.
It looks like only first layer have under-extrusion issues. I didn't find an option to increase flow rate on first layer only, so I increased overal flow and it's a bit better, but the filament is still being dragged around.
# ^ I wrote that yesterday, but decided to hold on a bit more and tweak settings and the machine a few times
I decided to fully take apart my extruder, reassembled it, cleaned glass, re-leveled printer and started printing Benchy a bit slower and few settings tweaked and ... I never got first layer so clean. All the layers seem to be better than ever, I hope the print will finish well.
Thanks for your help.