According to R' M. Willig of YU and Riverdale, the main theme of Rosh Hashanah night is coronation, while the theme of Rosh Hashanah morning is mishpat (judgment). All 6,000,000,000 people are judged sometime between midnight and chatzos (noon) on Rosh Hashanah morning. So you want to be doing something worthwhile, or at least not exposing oneself to the chance for sin, during that period.
While you're asleep, you're OK, you can't be doing anything wrong. While you're awake, though, you have to worry about temptation. If you're in shul, you're davening, you're thinking about Hashem, you're worrying about being judged - so God-willing, odds are that your microsecond of judgment will happen while you're doing the right thing.
So, since the hour of mishpat ends at chatzos, there's no good reason to finish davening before then, and expose the congregants to more temptation than necessary.