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The Mishna frequently mentions different opinions. Some opinions are generally rejected Halachicly (Rabbi Elazar, Beis Shammai). Why doesn't the Mishna mention who the Halacha follows (in general).
The Gemara is much more complex, with many arguments going back and forth, and yet also has very few final rulings.
Why is that?
Rav Yitchak Cohen from Yeshiva University explained this to me about 40 years ago when I asked him this question as a student in his class.
He said that the purpose of the Gemarah is not to always tell you the answers. It is the questions that are the most important. The purpose of the Gemarah is Torah Sheb'al Peh - the Oral law. In that sense, it mimicks the written Torah, itself. The written Torah generally does not explicitly state how to do something. One main reason for that technique is to have you delve into it so that you can ask questions and delve further into discovering the answers.
The Gemarah too, presents things in a question and answer style and offers various opinions with arguments and proofs and rebuttals. Why not just get to the point and tell you "do this"? Often the Gemarah doesn't know the answer which is what taiku is. But even if it did give you the final answer, which it frequently does, perhaps, more than you may realize, the purpose is really to appreciate the discussion and methodology and get you to ask the questions.
In short, Rav Cohen explained, that the questions are far more important than the answers. And, it's also a disciplinary lesson, Rav Cohen explained. It teaches you the discipline of learning to accept unanswered questions in life. (FYI - this is a tough challenge for the average person, but Rav Cohen taught me that learning Gemarah really does train you in this discipline!)
Your question is essentially answered in the Mishna in מסכת עדיות פרק א:
משנה ה:
ולמה מזכירין דברי היחיד בין המרובים, הואיל ואין הלכה אלא כדברי המרובין: שאם יראה בית דין את דברי היחיד, ויסמוך עליו, שאין בית דין יכול לבטל את דברי בית דין חברו, עד שיהא גדול ממנו בחכמה ובמניין. היה גדול ממנו בחכמה, אבל לא במניין, במניין, אבל לא בחכמה--אינו יכול לבטל את דברו: עד שיהא גדול ממנו בחכמה, ובמניין.
All opinions are mentioned, so that nobody can claim that they have a tradition of another opinion that should be taken into consideration.
There are rules how to Pasken from the Mishna and the Gemara. For example, we always follow the "multitude" רבים against an individual.
Many, if not most (all, possibly?) sugyot in the Gemara have a מסקנה - a conclusion - and that מסקנה becomes the Halacha. (Many of us have simply never been trained to conclude each section with its halachic ramification.)
When you find differences of opinions in Halacha, you will always be able to trace their origins back to different ways to learn the Sugya; if the opinion has no basis in the Gemara, how can it be a valid opinion?
