Now, its sort of hard to judge because we haven't gotten into Cav-El deeply enough to be sure, but from what we have seen no, I dont think Cavill's Superman would go that far off the deep end on his own. I could see him using lethal force when it wasn't the only option or otherwise being a much rougher, more dangerous Superman, but I think it would take a lot more than the death of a loved one for Clark to sell the world to Apokolips.
Now, mind control is another matter entirely. And honestly, I always figured the Knightmare scene in BvS was hinting at that sort of direction. We've seen a few stories where Darkseid mind controls Superman, so to see Clark leading a horde of parademons immediately brought that setup to mind, and the line from Clark about Lois dying or whatever it was, I took to mean that Lois' death is what made the crack in Clark's psyche that Darkseid slipped through.
And yes, I do believe that the Knightmare was supposed to be the endgame until the backlash hit and WB realized that just because Injustice was a popular video game, that doesnt mean it'd make a good movie.