"The one thing that stands in the way that the insignias presumably refer to assigned sectors of fleets would be "The Doomsday-Machine", IMHO.
Right from the beginning Spock states that they had charted the system in the previous year, so what's the Constellation's business in their "backyard", as it obviously has an insignia quite different from the one of the Enterprise?"
Not really, just another division is assigned to patrol the area that the "Enterprise" was assigned to last year. Similar to the rotation of navel deployments. One year a carrier say "Nimitz" is sent to the Persian Gulf and next year the "Stennis" is sent to the same area. I don't see that the "Constellation" is patrolling an area that the "Enterprise" previously had patrolled as an issue.
The thing more disturbing to me is in "The Omega Glory" when Kirk says that the "Exeter" had been patrolling in that area 6 months ago; he hadn't heard of any trouble. Really? No contact with Starfleet for six months and nobody was worried or sent a ship to investigate its disappearance? The "Enterprise" just by chance ran into the "Exeter" That is bizzare. Maybe they learned their lesson since they DID send the "Enterprise" out to search for the "Defiant" in the episode "The Tholian Web"