It would seem that the key is the Gemara (Yoma 70a) on this Mishnah. Most of these quotes can be found on Sefaria, and for the one that can’t, I’ve provided a link to HebrewBooks.
הרואה כהן גדול כו' לא מפני שאינו רשאי פשיטא מהו דתימא כדריש לקיש דאמר ריש לקיש אין מעבירין על המצות ומאי מצוה (משלי יד, כח) ברב עם הדרת מלך קמ"ל
“One who sees the High Priest...not because he’s not allowed.” This is obvious! [What reason should there be to prohibit going to just one?] What might you have thought [without the Mishnah saying this]? Like Reish Lakish, for Reish Lakish said, “We don’t pass over Mitzvos.” And what Mitzvah [would apply here]? “In a multitude of the people is the glory of the king.” [Therefore, everyone should go to the Temple to hear the Torah reading - Rashi.] It therefore comes to teach us [that it is permissible to go to either one].
The Gemara doesn’t explain why this teaching doesn’t apply, though. Rashi explains:
קמ"ל - דלאו מעבר הוא מאחר שאינו עסוק בה:
“It comes to teach us” that he is not passing over, since he is not directly involved in it.
According to this explanation, the teaching of Reish Lakish is completely inapplicable to the Mishnah - but the verse in Proverbs may still be relevant.
That still doesn’t help us, though, given how Rashi learns the preceding line:
ומאי מצוה - יש בראייתן יש כאן מצוה משום ברוב עם הדרת מלך:
“And what Mitzvah” is there in seeing them? There is here the Mitzvah of “in a multitude of the people is the glory of the king.”
Seeing them. It’s not clear to me whether “them” refers to the cow and goat being burned, or whether “them” refers to the Torah reading and the burning. It is clear, however, to the Orah v’Simcha (§73 on Rambam Hil. Avodas Yom HaKippurim):
דאיכא מצוה בראיית הקריאה ובראיית השריפה משום ברוב עם הדרת מלך וסד"א דאסור להניח מצוה זו ולילך לקראת מצוה אחרת קמ"ל דמותר מטעם שביאר רש"י שם כיון שאינו עסיק בה
For there is a mitzvah is seeing the reading and in seeing the burning, because of “in a multitude of the people there is the glory of the king,” and I might have thought that it’s forbidden to leave this mitzvah and go to another one. The Mishnah comes to teach us [that it’s fine], for the reason that Rashi explains there, that since he’s not involved in it.
It’s clear from his ending that he’s working in Rashi’s understanding the Gemara; therefore, he learns Rashi according to the second approach I posed above. (After the above quote, he poses a problem with fitting this approach into the Rambam, but ultimately he infers that this is indeed the Rambam’s position.)1
Steinsaltz takes a hard stance on it referring specifically to the Torah reading:
ומאי [ומה] מצוה יש כאן בשמיעת הקריאה מפי הכהן הגדול
“And what mitzvah” is there here in hearing the reading from the mouth of the High Priest? [...]
According to this approach, it would seem that one should go to hear the Torah being read, along the same line of logic as above.
The Ritva has a completely different understanding of the question Gemara that may shed a different light on its conclusion.
לא מפני שאינו רשאי פשיט׳. כלומר פשיטא דרשאי להניח קריאת התורה וללכת לראות שריפת הפרים דאע״גב דאמרינן ועוזבי ה׳ יכלו זה המניח ספר תורה ויוצא הרי רשאי לצאת בכאן בין פרשת אחרי מות לפרשת בעשור שאפילו בין גברא לגברא מותר וכ״ש בין פרשה לפרשה ואע״ג דתרווייהו בחד גברא:
“Not because it’s impermissible - that’s obvious!” That is to say: it’s obvious that you can leave the reading of the Torah to go see the cows being burned, for even though we say “‘Those who abandon Hashem will be destroyed’ - this refers to one who leaves a Sefer Torah and goes out,” here it’s permissible to leave, between Parshas Acharei Mos and Parshas Asor, for even between two people it’s permitted, and certainly between two sections, even if they are read by the same person.
The Ritva does not address the conclusion of the Gemara. However, notice that, unlike Rashi, he learns the question as leaving the Torah reading to watch the cows being burned, rather than initially choosing to go rather than stay and listen. It would seem from that understanding of the Gemara that it’s assumed that you’ll go and listen, and Reish Lakish is only brought up as a potential reason to stay.
Now, “in a multitude of the people” would seem to indicate that whatever your interpretation of the Gemara’s initial position, that would be the preferable thing to see (not to say that it’s required, as per the Gemara’s conclusion). Therefore:
According to the Orah v’Simchah’s reading of Rashi and the Rambam, it would seem to not matter which one, though once you choose, you should stick with it.
According to Steinsaltz, it would seem that one should preferably go to hear the Torah reading. For different reasons, this would seem to be the Ritva’s approach as well.
1He later answers your parenthetical question about why it says “seeing” and not “hearing”:
וצ"ע והא השומע כעונה וא"כ זה הרואה כה"ג קורא הא הוי בכלל עוסק במצוות דשומע כעונה והרי זה עוסק בתלמוד תורה, ואפשר לומר דאיה"נ אלו השומעים אסור להם לצאת ורק הרואין אמרו, שהיו שם כל העם ולא כולם היו יכולים לשמוע.
And [Rashi’s explanation] requires looking into, for one who hears is like one who says, and therefore, this one who sees the High Priest reading should be considered being involved in a Mitzvah, for, due to “one who hears us like one who says,” he is toiling in Torah! Perhaps you can say that, indeed, those who hear him may not leave; only those who “see” did they say [can leave]. For the whole people is there, and not everyone could hear.
He notes at the end that it’s obvious from this whole discussion that there’s no obligation to hear the Torah reading, or else there would be no possibility of anyone watching the burning.
