The funniest thing about the nonsensical claim
"What happens if you bring death to death? You bring life!"
is that the Doctor doesn't even believe it himself
I mean, by bringing Sutekh - the personification of Death - to all those dead worlds, the worlds come back to life? Ok, i think it's silly, but if you say so...
BUT THEN
"If you're Death, then I must represent Life! [...] You turned me into this. I am the one who brings death."
So the doctor literally SAYS, explicitly, that he is the one who brings death to Death by killing Sutekh.
So why is he sad about that, huh? By his OWN logic, he just brought life! He brought death to Death!
It just doesn't make sense man...