Chana's Prayer and Disruptive Davening on Rosh Hashana

Throughout the year when we pray, the centerpiece of our tefilot is the Shemoneh Esrei, the Silent Devotion. We stand in silent prayer as we praise God, ask Him for things we and the Jewish people desperately need and thank Him for all that He has done and continues to do for us on a daily basis. It might then come as surprise that Rabbi Yosef Karo codifies in his Shulchan Aruch (Orach Chaim 582:9) the following halacha:

אף על פי שכל ימות השנה מתפללים בלחש, בראש השנה ויום הכפורים נוהגין לומר בקול רם. ולהטעות לא חיישינן, כיון שמצויים בידם מחזורים
Even though throughout the rest of the year we pray silently, on Rosh Hashana and Yom Kippur it is customary to utter [the prayers] aloud. And we are not concerned that this will cause others to make a mistake because they have machzorim (prayer books) in their hands.

It is apparent from Rabbi Karo's ruling that in his day (16th century) people did not regularly daven with siddurim. It was only during the High Holidays, when people were unfamiliar with the prayers that it was customary for everyone to have their own prayer book; people were more susceptible to make mistakes in their tefilot when they prayed by heart so it was prohibited to daven out loud, lest you break another person's concentration. However on Rosh Hashana and Yom Kippur when everyone had machzorim, people would not lose their place in tefila if they heard someone else praying out loud.
Rabbi Yisrael Meir Kagan in his Mishna Berura (582:20) explains why one would raise their voice especially on Rosh Hashana and Yom Kippur,

מפני שעי"ז יוכלו להתפלל יותר בכונה ומ"מ לא יגביה קולו יותר מדאי
For through this (raising one's voice) one can daven with more kavanah (intent). Nevertheless, one should not raise their voice more than necessary.

Nowadays even when people use siddurim year round the custom is to never raise one's voice during the silent Amidah because it is seen as a distraction and perhaps even as a display of arrogance.
One could make a strong case for praying silently, especially on Rosh Hashana. On the first day of Rosh Hashana we read of the birth of Shmuel. His mother, Chana, barren and desperate for a child turned to God in prayer. The Navi tells us (Shmuel I 1:13),

וחנה היא מדברת על לבה רק שפתיה נעות וקולה לא ישמע
Chana was speaking to her heart, only her lips moved, but her voice was not heard...

The Gemara (Brachot 31a) learns the following halacha from this pasuk,

רק שפתיה נעות - מכאן למתפלל שיחתוך בשפתיו. וקולה לא ישמע - מכאן, שאסור להגביה קולו בתפלתו.

'Only her lips moved' - from here we learn that one who prays should mouth the words with their lips.
'But her voice was not heard' - from here we learn that it is prohibited to raise one's voice during their tefila

These halachot refer to specifically to the recitation of the Shemoneh Esrei. We learn these laws, along with many others, from Chana's successful prayer. Rabbi Kagan writes in the Mishna Berura (584:6) that we read about Chana on Rosh Hashana because Hashem remembered her on that day and enabled her to conceive. Perhaps this is the reason why the custom has remained for people not to raise their voices during tefila even on Rosh Hashana, for it would seem inappropriate to deviate from the rules we learn about prayer from Chana on the very day her prayers were answered.