The Moon Presence Doesn't Want Us To Hunt The Great Ones.

I don't think the Moon Presence really cares whether we hunt the Great Ones or not. After all, the Ebrietas and Amygdala fights are optional. The only otherworldly beings we kill are Rom and the Wet Nurse. Even then, we only kill them to fulfill our contract. 

The contract, in my opinion, involves freeing Mergo from the Nightmare of Mensis. Why do I think that? Firstly, we don't lay a finger on the infant. We simply let him dissipate after besting the Wet Nurse. Secondly, some say that Mergo will laugh before leaving the nightmare, if we play the lullaby from the music box for him. Him laughing makes me doubt that our contract would involve harming Mergo. Finally, Queen Yharnam seems to be thankful to us for freeing Mergo from the nightmare. I'm pretty sure if we came to the nightmare to slaughter Yharnam's child, she would not be so welcoming toward our actions.

I do think some institutions want to hunt the Great Ones. The School of Mensis is one of them.

A note can be found on the second floor of the lecture hall saying "hunt the great ones". Yet it's unclear who wrote it. Perhaps it was written by someone who believed strongly in the teachings of Mensis.

Moreover, Damian of Mensis can be summoned for both, the Rom and Ebrietas fights. Which could further point to the School of Mensis disliking the Great Ones, in some capacity.

And keep in mind, the Moon Presence could probably easily propagate its plan to hunt the Great Ones though Gehrman. Yet Gehrman simply directs us where to go until we reach Mergo. He doesn't specifically tell us that we must kill creatures that may come from the Sky or the Sea.

Of course the question then becomes: Why does Mensis dislike the Great Ones? Perhaps because some of the great ones (e.g., Rom and Ebrietas) did not appreciate whatever Mensis was trying to achieve, and attempted to stop them. After all, Rom does block the ritual of mensis. And Ebrietas seems to spend her time at a place which has a statue reminiscent of Rom. 

I'm still thinking about this whole thing, hopefully I'll have something more coherent at some point!

Thanks for reading!