Most of the commentators agree with the premise you state: there is no need to study the Mishna alone. The Mishna clearly held no independent standing in the eyes of Rashi as in his commentary he would not flesh out the concepts presented in the Mishna, but rather said “it is explained in the Gemara,” ("בגמרא מפרש"); in other words, the concepts and complexities in the Mishna will be addressed in the Gemara, and thus the reader should just look ahead to the Gemara’s discussion to answer his questions. Throughout the era of the Geonim, those commentaries that were written on Mishna were limited in scope and purpose focusing almost exclusively with material with respect to which no Talmud Bavli was written.
Then came the famous commentary of the Rambam on the Mishna. In that era, Mishna study was not a normal independent pursuit and Mishna commentary not typical. So what motivated the Rambam in this regard? In his introduction, although he does not expressly state his motivation for writing the commentary, he does address four benefits that he believes the work will serve:
האחת שאנו מלמדים פירוש המשנה על נכון ובאור מליה שאילו שאלת לגדול
שבגאונים בעד פירוש הלכה מן המשנה לא יכול לומר בה דבר אלא אם ידע על פה
תלמוד ההלכה ההיא או יאמר לך אראה מה שיש בה מן הדברים בתלמוד ואין
ביכולת אדם לדעת על פה כל התלמוד וכל שכן בהיות הלכה אחת מן המשנה מפורשת
בארבעה עלים וחמשה כי יבא דבר על דבר וטענות וקושיות ותשובות עד שלא ידע
לבחור הברור מפירוש המשנה ההיא אלא אדם בקי בעיון.
(1) The one is that we teach the correct explanation of the Mishnah
and the meaning of its words. As if you were to ask the greatest of
the Geonim about the explanation of a law of the Mishnah, he would not
be able to tell you anything unless the knew the Talmud of that law by
heart; or he would say to you, "I will see that which is [written]
about it in the words of the Talmud." As it is not possible for a man
– unless he is an expert in reseach – to know all of the Talmud by
heart; and all the more so in that one law of the Mishnah is explained
in four or five pages – since one matter leads to another and [to]
arguments and objections and rebuttals – to the point that he will not
know [how] to choose the concise [conclusion] from the [Talmud's]
explanation of the Mishnah.
והנה מה יעשה בהלכה אחת כי לא ישלם פירושה ופסקי הדין בה אלא משתי
המסכתות או שלשה.
And behold, what will he do about a law the explanation and legal
decision of which is only completed in two or three tractates?
והתועלת השנית הם הפסקים שאומר לך בפירוש כל הלכה על דעת מי נפסקה ההלכה.
(2) And the second purpose is the decisions, since I will tell you in the
explanation of each law, according to [which of the various opinions]
is the law.
והתועלת השלישי להיות כמבוא לכל המתחיל בעיון החכמה ויקבץ לו ממנו דרך
בדיקת הדבור וביאורו ויהיה אז כמי שכלל בדעתו כל התלמוד ויהיה לו לעזר
גדול בכל התלמוד.
(3) And the third goal is to be like an introduction to anyone who is a
beginner to the research of wisdom. And he can gather from it a path
to check speech and its clarification, and he will then be like one
who encompasses all of the Talmud in his mind. And it will be a great
assistance to him in all of the Talmud.
והתועלת הרביעית שיהיה למזכרת למי שיקרא וידע ויהיה כל מה שקרא נצב בין
עיניו תמיד ותהיה משנתו ותלמודו סדורה על פיו.
(4) And the fourth goal is that it be a mnemonic for the one who [studied]
and knew [the Talmud, such that] all that he read will always be
present in front of his eyes, and his [knowledge of] Mishnah and
Talmud will be arranged in his mouth.
So the Rambam in a way answers your question: "why is the Mishna a separate subject of study?"
Also, in his commentary to Mikvaot, the Rambam cites a debate among certain commentators as to whether a specific kind of mikva is kosher and wrote about how they were needlessly debating a matter that could be found explicitly addressed in the Mishna and notes that what caused this in his view was the neglect of Mishna study.
