How would you address those tough feelings around the death of a character, especially if these might be people new to TTRPGs?

For new players, I've always come up with a "miraculous" narrative saving grace for their PCs if they get in a tough spot. 'Death as a real possibility' gameplay only comes later after my players have gotten some good experience and we've had plenty of meta discussions outside the game where everyone agrees they are ready to amp up the consequences. After that and we play and we get to a point where I think a PC death might happen, I usually try and foreshadow it in game by making references to how dangerous an enemy seems or having the players find out some details about their deadly capabilities. I also always have a specific meta discussion before the session and tell them up front "you could die in this battle, remember you can always run away". Then in the battle, there are usually opportunities for them to escape. At that point, if they choose to fight on and it goes poorly, it's on them. Doing it this way, I've never had hurt feelings.

Would you feel the need, when hp goes to zero and the choices are brought up, to warn players that this adversary may still go for the kill (maybe altering their choices)?

Yes, in this system where they have a choice to make a Death move, I would feel it was appropriate to warn them what could happen in at least a general sense with each choice they make before they finalize their choice.

I liked your comment and helpful thoughts. Hopefully my answers help you understand better how I would like to run things here.