|You know it makes sense|
Alternative time-lines do not appear as such in TOS, unless you count the Mirror Universe, which I don't. In later Star Trek there are quite a lot of examples. However, TOS does contain alternative Earths. They are simply other planets, reached by space travel. But these planets had been much the same as Earth until something went differently.
There is seldom much attempt to explain why these planets exist. If you're into the hard SF explanations, there is an argument that if the universe is infinite all possible worlds will exist. (This was explored in Arthur C. Clarke's neglected story "The Other Tiger".) But you wouldn't expect a whole lot of them close by. However, the idea of a planet like Earth was easier for the audience at that time than the time-lines version.
One example is "Bread and Circuses". The planet is a vision of a world where the Roman Empire never fell. Incidentally, it's not an exact replica: the land-masses are different. But the important things seem to be the same. It has reached the twentieth-century level, paralleling the world of the viewers rather than that of Captain Kirk. Some things are obvious, such as the arena being televised. But there has been historical change. Slavery has evolved into a softer form, and slaves now have medical care and old age pensions. A television report expresses bewilderment at resistance by well-treated slaves. This is an interesting aspect, which can be taken as a comment on contemporary civil rights issues, though perhaps also more generally on the widespread belief of the powerful that the poor and oppressed ought to be grateful. (The universal medical coverage of these slaves is rather ironic in view of the actual situation of the United States in the present day.)
The episode is rather unusual in Star Trek for its explicit reference to Christianity. This exists in the parallel Roman Empire but is only just starting to take off. The crew had not understood this till the end because they thought people were talking about worshipping the Sun, not the Son, due to the parallel Earth speaking parallel English. Spock predicts that Christianity will replace the cruelty of Rome, and Kirk reflects it would be wonderful to see the advent of Christianity all over again. Star Trek is often said to reflect a secular and even atheist view of the future, but that is definitely too simple. (Secularism is at its height in TNG.) Incidentally, the episode doesn't actually say that Christianity is true: the story makes sense even if it is just a (parallel) good thing.
Another one where the Alternative Earth is necessary is "The Omega Glory" (TOS), which it has to be said is a weak episode. Here there was a Third World War presumably in the Parallel Twentieth Century, some time ago, leading to the destruction of civilization. The descendants of the Americans are carrying around the United States Constitution but have forgotten what it means, reciting garbled and meaningless words. So Kirk has to tell them. This seems to be an allegory of the ideals of the Constitution being recited with reverence but not applied. There is also a subplot about a renegade Starfleet captain, which gives the episode one really good scene.
Incidentally, "Bread and Circuses" and "The Omega Glory", which are fairly close in sequence, both involve a renegade captain. The interesting thing is that in "The Omega Glory" it's an admired Starfleet captain, who goes completely off the rails but justifies his actions to himself. In "Bread and Circuses" it's a merchant-ship captain, who failed his Starfleet training. He too fails his duty, but he is aware of it, and at the end redeems himself by sacrificing his life. Taken together they suggest a moral.
In other cases it's not so clear why Parallel Earth is necessary. "Miri" (TOS) is about a world where there was a disastrous scientific experiment in the Parallel 1960s (several hundred years ago) aimed at extending life. It seems that it did so, but only for children, who age very slowly: at puberty they go violently crazy and die, and since all the adults had died when the infection got out there are only children left. This planet is said to be an exact duplicate of Earth. Since the Enterprise visits alien planets which look fairly like Earth, it's not obvious why it was necessary to have a full Parallel Earth here, but perhaps it's because "Miri" comes fairly early in Star Trek.
Some other episodes have planets that look like Alternative Earths but aren't really. In "A Piece of the Action" (TOS), for example, the society looks like a world of 1920s gangsters, but only because it is a deliberate imitation of them. In "Patterns of Force" (TOS), which I regard as probably the worst episode in TOS, a human historian has created a Nazi society. In cases like that, the Earth-like aspect is a modification of what would otherwise have been an alien world.