The simple answer is that the entire discussion about the moon appearing like the sun is not literal. It is simply a teaching aid, and a model that is referenced in another area of the Torah.
That if this idea of using allegorical models and paradigms is confusing, think about how chemistry is taught in school. Complex discussions of molecular structure, for example, are often presented through the use of simplified physical models that can be understood by everyone.
The simplified models presented in the five books of Moshe are even known to a child of five years old like is said by Yehuda ben Teima in Avot 5:21. And one of the requirements of learning Mishnah is proper comprehension and understanding.
This idea, that this particular detail is allegorical, is emphasized by the commentaries to the very Mishnah you link to, for example the Bartenura who says in regard to the phrase ”one time the luminary called the Moon (הלבנה) rose etc.:
לאו ביוה״כ קאמר, דא״א לעלות מאור הלבנה סמוך לשחר ביוה״כ שהוא בשליש החודש
This isn’t speaking about Yom HaKippurim. That it is impossible for the luminary called HaLevanah* to rise close to morning on Yom HaKippurim, which is in the first third of the month. Rather, in the final third of the month is when the moon rises close to dawn. And this event is an error (See Mishnah Brachot 2:3) and causes us to apprehend (meaning to grasp with understanding), lest in regard to Yom HaKippurim, other errors will occur. We are required to act in a similar fashion in regard to everything.
One of the points brought out in Brachot 2:3 is that when you see an error, go back to the beginning. In context, look to the source within the Torah for the Avodah of Teshuvah, which appears in parshat Bereshit as cited below.
That it (this reference to the moon being confused with sun) is not speaking about Yom HaKippurim. That it is impossible for the light of the (full) moon to appear at Yom HaKippurim which is in the first third of the month...
And also as Rambam emphasizes:
ידוע כי הלבנה אינה עולה מן המזרח אלא אחר חצות החדש.
And it is known that the moon doesn't ascend from the east unless it is after half of the month... (and Yom HaKippurim is always on the 10th of Tishrei in the first 1/3 of the month)
And in the context of the first chapter of Mishnah Yoma which begins this discussion and also references the first 7 days (before Yom HaKippurim) and the subject of husband and wife (all of which alludes to the allegorical reference model, namely the seven days of creation and the creation of Adam, the husband, and the wife, Chava), it points to the Torah source which is the paradigm from which these laws are based.
And this is also asked in the commentary of Yachin to the Mishnah you reference which questions why it was necessary:
לעלות לגג לראות, ולא הסתפקו בראי' פשוטה למזרח
To ascend to the roof (another allusion to this spiritual/Heavenly/Angelic paradigm from the 7 days of creation, meaning the 6 days of building [that roof, גג which is gematria 6] and Shabbat Kodesh. Like is brought in Talmud, the 6000 years of this world are based upon the paradigm established in the 6 Days of Building, building בנין which is also the language used in the formation of Chava, Adam’s wife.). Wouldn't it be sufficient to glance to the east like the simple meaning of the words? But Yachin states that it is necessary to understand the concept properly.
That after the creation of Adam HaRishon, he was separated into two individuals, the husband, Adam, who would be the Mashpia, and Chava, the wife, who would be the Mekabel.
And this process was based upon the earlier paradigm which G-d established in regard to the appearance of the single light on the first day of creation and then the subsequent creation of the two (equal) great sources of that single light within the creation itself, like is stated in Bereshit 1:14-16.
(That this follows the general Torah concept that there are three partners in a Jewish marriage, husband, wife and G-d [who hovers over the marriage bed] like is implied from the words איש and אשה which are formed from two equal parts of fire or light (אש) and G-d (יה) who unites them to be a manifestation of oneness (one flesh).
And Torah emphasizes in midrash to Bereshit 1:14-16 that as a consequence of the insistence of the moon, meaning that the moon said in order to proclaim G-d's oneness as King (to avoid the confusion that there might be two G-ds, two Kings) requires that one of us must be reduced in stature (temporarily) and serve as a Mekabel, G-d reduced the Moon. And that appearance, that the sun and moon look obviously different, is what is seen during the entire period of exile.
But the Torah teaches that ultimately, when we have returned to G-d in teshuvah, another reference to the concept of Yom HaKippurim, meaning during the final redemption, the moon will return to its true, original state, like is said in the Santification of the Moon prayer.
And it is in that context, meaning in the context of the ultimate point of Yom HaKippurim (that all Israel will return to their Father in Heaven, and the reduced appearance/condition of the moon will be returned to its fullness) that the appearance of the moon could be confused with the appearance of the sun.