Unpopular Opinion: I think that the Anime handled Yuugo better that the VN
In both the game and the show, it's explained that once the rounders served their purpose, they get killed so that there are no loose ends. If you screw up, they go after your loved ones.
Yuugo did want Moeka to live, but it means that Nae would end up getting hurt (most likely killed at that), so it actually makes more sense that she died by his hand after all, as he was tasked to kill her in the first place. Remember, failure = your family and friends are in danger.
I think that the D-Mail stuff delaying Mayuri's death did exactly the same to Moeka, changing the way she died (her suicide). Her helping Okabe and exposing Yuugo gave him the necessary push at this point in the story, but eventually he always intended to kill her in order to protect Nae by doing his job. Moeka just couldn't stand being ignored anymore and she broke down.
As I said before, he did want her to live, so regardless of the outcome, he was going to shoot himself in the end out of guilt, murder or not. The reason why it took him too long for Moeka (which lead to her suicide), is because he kept hesitating as he did not want to do it and that he was biding his time, spending as much time as possible with Nae.
Edit: Thank you to all the Anime fans in here for trying to keep my post upvoted haha