I think people just assumed Mergo died because of the "Every Great One loses its child" and the blood on Yharnam's dress. But you're right, it doesn't explicitly say Mergo's dead.
We also don't know if Arianna's baby would die anyway, if it would, Oedon is kind of an idiot. Every red moon he does the same thing, it never works, and he keeps doing it anyway.
Well, assuming I didn't miss any major/explicit lore regarding Mergo's current status, then I feel like it's fair to assume Mergo is actually alive. This makes more sense to me because your goal, as a hunter tied to the dream, seems to be to kill the newborn Great One. All Great Ones do lose their children, but that doesn't mean ALL of their children are lost. If I remember correctly, it's worded in such a way where it says they lose their children, and THEN seek a surrogate. This seems to indicate that the children they have using surrogates can and often are successful, with Mergo being the most recent case.
So far, this is how I can make sense of the Moon Presence's motivation. It seems like the MP does not want competition in the way of new Great Ones, so it sets up the dream to guide a hunter to kill it during the hunt. This also explains why, if you've ingested 3 umbilical cords, the MP jumps back from you (maybe in fear or disgust?) and begins to fight. It senses that you're now a threat, and so tries to dispose of you.
That is a good question. After defeating Rom, I believe we are instructed to "slay the nightmare newborn" (or something like that), which would make me think that Mergo was just born. But that can't be right if Yharnam gave birth to Mergo however many years ago.
One thing I can think of is that maybe death is a very different experience for Great Ones, and perhaps it's not really permanent. The goal of the hunt (at least for hunter's tied to the dream) seems to be to hunt down and kill the newborn. Maybe the newborn needs to be repeatedly "killed", in order to stop it from growing up into a fully formed Great One? Every night of the hunt, the Moon Presence picks a special hunter to carry out this task. Before it was Djura and Eileen, and now it's your turn. (Also, as a side note, Mergo being described as a newborn might not mean much in terms of time, since who knows how long a Great One baby takes to "grow up"?)
I remember reading somewhere that if you play the music box in the Wet Nurse's boss room after killing her/Mergo, then you can hear a baby laughing. I haven't had a chance to test this since I'm still in the beginning of NG+, but if it's true then it might support the idea that even though you 'killed' the newborn, it still exists in some form.
Another thought I just had that's kinda out there, but what if the reason the Great Ones always lose their children is because other Great Ones keep killing them? Maybe they're all jealous of each other, so they pretty much live in a cycle where they're killing each others babies over and over again?
It's interesting to think that the hunters, the scourge of beasts, and everything the Pthumerians/Yharnamites have gone through could just be insignificant pieces in a game played by the Great Ones.
Also, another thing that just came to mind, but could the birth of a newborn Great One, or in Mergo's case a possibly "reborn" Great One, somehow be a trigger for the hunt? It could be tied to the increase in beasts or something to do with the ritual reaching some kind of end... I dunno, I'm just writing down a random stream of consciousness at this point... Trying to tie everything to the MP and it's motivations. I probably need to revisit the item descriptions regarding the hunt.
-----(LAST EDIT, I SWEAR)-----
Completely unrelated thought I just had, but I wanna write it down here for reference. I've always been a bit confused regarding some theories that indicate the world we're in (Yharnam) is some kind of dream world or alternate dimension. The guy at the beginning of the game who begins your first blood transfusion seems to indicate it as being some kind of nightmare. I was thinking that maybe what this means is that people who use the special blood of Yharnam enter into the sphere of influence of the Great Ones. They now become somewhat aware of them, and are part of their world, thus entering a type of "nightmare". It could also mean that you literally enter an alternate nightmare world. I've heard people mention parallel dimensions/dreams that the Great Ones like to dwell in, so maybe all those who use the blood are slipped into this hidden dimension of the Great Ones which exists on top of the "normal" world. In this way, we're still in the "real" world of Yharnam, but on the same plane of existence as the Great Ones as well.