I don't know what happened but with recent update in SOME way my problem was solved.
The counter was showing 0/6 at any point whenever I've had built or not those 6 furnaces or drills. As was as I know, none of the vanilla buildings are not changed in my mod pack. As well as those drills and furnaces.
That brings us to the last point about overhaul. Nope, I don't use such gigantic mods besides cosmetics, QOL-mods and those that ADD new recipies and buildings but much later in the game (like Aquilo Lithium Batteries or Atomic Forge), but do not change main vanilla entities. Maybe it is affected by the Mythical Quality mod (adds 6th grade), but that way EVERY other entity and item should be changed which is not the case.
What's interesting, other quests with build condition (like assemblers) are passed successfully as well.
So, back to the update. I've updated your mod and THAT particular quest passed through successfully and many others as well! BUT! I didn't actually met the prerequisites!!! So I don't know how this quest was passed even though I've never crafted burner drills (beside the one from the start). The game solved several next quests correctly (science packs, assemblers, etc.) But yet it stumbeled again with "black gold" where I need to place 4 pumpjacks and a refinery. Althought other tasks (10k gas and research) are done. Of course I already have them and building new ones do not solve this problem.
UPD: I DID single out other mods that may interfere. In my case after I turned off "Winter is coming" by valertron I've progressed to a quest with drones which I didn't complete for real and it seems correct.