It’s possible that the red-shirted members would have gotten aggressive over time; they simply died so frequently that it was never much of an issue.
Judging by Aaron Harberts’ comments to EW, there will be impactful conflict among the ranks of the starship Discovery.
While the early years of Star Trek under Gene Roddenberry did feature conflict, it was usually between the crew of the Enterprise and the various alien species they encountered on their travels.
Sure, Spock and Bones would disagree fundamentally sometimes, and there was the occasional case of mind control and/or body swapping that led to in-fighting on board, but the crew members were usually pretty cool with each other.