I agree with Nerys that there's no reason to believe we saw the entirety of the mission.
And it's not unrealistic to think a single disastrous mission could kill 20 redshirts in one go. We likely saw a very small glimpse of Enterprise's variety of missions, especially given that the show had obvious real-world limitations.
I wonder if anyone' ever attempted to keep track of redshirt deaths in the novels and comics? I remember He Who Shall Not Be Named kept a list of 430 crewman which included all the episodes and novels up until about 1998 or therabouts - but there have been a LOT of TOS tie-ins since.
I was pleased with NuTrek and Olson's outing. He did Star Trek redshirts well.
Last Words: Yeehaw!
I loved the look of terror (and accompanying music sting) when Chekov was told "go put on a red shirt" in Into Darkness
Perhaps, since Kirk hadn't yet lost a crewman at that point, one can look past the fourth wall gag and speculate that the "redshirt curse" began with another, older ship?