Well... taken by itself there's nothing wrong in this part (though maybe the actual deaths could be longer and have more details), but it doesn't feel like it belongs with the premise set up in the first part. You say it yourself that the point of this third part is to give the characters reason why they want to die, but that doesn't fit well with how the first part established they don't need a reason beyond Hayate telling them to. The first part didn't explain why do the characters go along with dying even if they feel sad about it and why did Hayate order them to die in the first place. But as long as it was the whole story it could be written off as hentai logic. But now the continuation not following the same logic brings those questions back.
Anyway, since you say the next part will be more like the first, I'm looking forward to it.