It is six weeks, later, the ship is refitted (now sporting regular series appearance rather than the pilot episodes stylings). Crew rotations are done. And the ship is back out to the frontier, with a schedule of planetary surveys ahead.
Then they get a message from Command. The USS T'Lera (a Vulcan explorer) detected a massive energy signature from the unexplored J400 system (near the Klingon border). The Constitution is being sent to investigate.
7 planets, #2 is Class M. No moon, but largely Earth-like.
Sensors show no high energy readings from the planet. But there was recent heavy meteor activity. There is evidence, that once collected and analyzed, shows a large meteor was on collision course (an Extinction Level Event). A force directed from the planet caused it to break up, altering its course. This also caused high ionization over the populated area of the planet (heavy rain and weather).
The indigenous population are human, based on atmospheric pollution, approaching the industrial revolution. Older structures bear resemblance to the Mississippian Mound Builders (a culture that vanished around 1400AD).
Sensors show one such building has a a concentration of an otherwise alien mineral - the same one used in the construction of the obelisk on Luna.
Probes can provide data for the Universal Translators and Material Fabricators (clothing). Then a Landing Party can beam down near the city to investigate.
The People of Caddoan have a legend. Those who cling to the old superstitions believe that long ago the "Sky Fathers" brought them here from a different land. To please the Sky Fathers, the Caddoans sought to learn, grow, and become pleasing for their return. And eventually achieved social and technological levels comparable to 18th century Terra.
Then a new philosophy was born. It declared the sky Fathers to be a myth, and demanded society focus on logic and reason instead.
At first they were ridiculed, the Obelisk at the heart of the Capital Building was proof. Later they were tolerated, since they advocated noble pursuits. Then they gained positions as educators, and subsequent generations were indoctrinated to support their beliefs.
150 years later, they have become the law of the land. Dissenters are considered dangerous, and treated accordingly, especially if they tried to form isolated communities of like minded individuals. And any decline in scientific progress in the last few generations is clearly the cause of fanatics working against the greater good, the need for resources to oppose them, etc. Not because of a sense of entitlement, and lack of striving on the part of the general population.
And then there was a terrorist attack on the capital, killing much of the government, with a weapon that shot light into the sky. Police and military forces are hunting the Religious Extremists who caused this. (While denying the existence of the Obelisk that was in a hidden courtyard underneath the government chambers - built a few decades ago).
The natives are, of course, highly suspicious of strangers. And the Landing Party are very strange.
Eventually, they can encounter a hidden group of the religious group. Their old religious text hides secrets they are unaware of, which will prove important to the party. They are peaceful, and know of no one in their sect that would have taken such an action. Nor of any explosive that would cause such a beam of light.
The ship receives a distress signal from a nearby civilian ship, and must go to answer it. No sooner have they left, then the one's responsible, the D-10 Command Cruiser Kagga's Crown (The Constitution's nemesis from since the days of the Four Year's War) enter orbit. A Klingon Landing Party beams in openly to the Capital Building (not burdened by the Prime Directive), using a Marine squad to secure it, while the ship's Sensor Master examines the artifact.
The Starfleet Landing Party must act to drive off the Klingons, and prevent them from gaining the secrets of a massive Tractor/Pressor Beam more powerful than anything in either nation's navies.
Fortunately the old religious text had data on how to access the obelisk (musical notes sounding much like - but not obviously like - a communicator's activation chime followed by "Kirk to Enterprise").
The party will not be able to fully interpret what they are looking at inside, but can make guesses to target power it up, and target the Klingon ship. And their own vessel will come warping back in once they learn they were tricked. The Klingons will back off, and leave, rather than risk destruction. And the Obelisk can protect itself. Unfortunately, it can provide little knowledge about the Sky Fathers.
Nor can the Landing Party do much for the persecuted natives. But exposure to the existence of hostile, advanced, aliens will make the general public begin questioning their beliefs. The Constitution continues her mission, and the Kagga's Crown is out there looking to even the score again. (And War looms ever closer).
Obviously, the situation on Caddoan is reflective of the way religion and secularism have developed in the last century. Whether you believe or not, and despite some terrible things done in the name of Christianity (among others) religion was for a long time the gateway to important discoveries. Even Darwin set out to understand Creation when he proposed Evolution. (Personally, I obviously do not find faith and science to be mutually exclusive). The situation here reflects where we could be in a generation or two if extremism prevails, rather than seeking middle ground.
I could go on, but the point in a Trek story isn't to provide what the author thinks is the "only correct solution", but to provide impetus for discussion. To let the viewers/readers look for the middle ground.