|Woman putting hands through openings to adjust ohashori. Image from here.|
Underarms of woman's and men's kimono, respectively. Image from here.
So despite how often it's mentioned, I'm not convinced that the uses listed above are the main reason why the armpit openings persist in female kimono. They are more subsequent uses.
This hole is, surprisingly, older than the modern kimono itself. The openings can be seen on the layers of the Juunihitoe. I suspect that the opening came along with Hanfu (Chinese clothing) when Japan began importing and adopting Chinese culture during the Heian period. (But, unfortunately for me, I do not know enough about Hanfu yet to find any examples but rest assured I will investigate).
Below is a replica from the 12th century and you can see the edges of the miyatsuguchi bowing out behind her arms.
Images from here.
Below shows the costume of a "Courtier in regular court dress, hitatare, like warrior, in Muromachi era". You can see miyatsuguchi behind his large draping sleeves as well.
Images from here.
Since it appears that men did have them at one point, the opening seems to be mostly characteristic of garments with large draping sleeves, specifically, kakusode 角袖"square sleeves" (*), rather than gender.
I believe the function of the miyatsuguchi is to allow for movement on garments with square sleeves that are meant to drape. On modern female kimono, being that the obi is so wide, arm movement would be constricted if the body sides and sleeve edges were sewn closed. This is because the back edge of the opening will move separately from the front edge, thus absorbing displacement during movement.
Liza Dalby mentions this in her book "Kimono: Fashioning Culture";
"If these sleeves had been sewn to the sides of the kosode, free movement of the arms would have been severely inhibited." (p.44)With historical clothing, I can imagine the opening would have also helped the many layers of kimono remain flat. It's basically the kimono equivalent to the function of the yoke on the back of a man's blouse. The pleats sewn below the yoke allow for extended arm movement.
Man's shirt showing shoulder yoke with pleats. Image from here
Since the sleeves on men's kimono no longer drape significantly below the obi, there isn't a need to keep the sleeve edges open. Arm movement isn't a problem because it has all this unbounded fabric at the sleeve and torso at its disposal.
Rear view of male costumes in Aoi festival ~ Japan 2015. Image from here.
The miyatsuguchi isn't necessary for everything. In this outfit below, there are no openings in the outer garment; instead the entire length of the sleeve is sewn to the body. This of course prohibits the use of an obi around it but clearly it wasn't intended to be used with one anyways. (I guess ventilation wasn't an issue in this case either).
"Woman of the upper warrior class in formal costume." The Costume Museum.
"Katsura-me or peasant woman of Katsura village selling food in Kyoto". The Costume Museum.
When the kosode first emerged, it was tied with only a cord and the sleeves were as long as the length between the shoulder and the hip were the obi was tied. Therefore, the sleeves were sewn closed against the body. Once the sleeves began to lengthen beyond the obi again around the 16th century did the miyatsuguchi reemerge, for both men and woman at the time.
In all honesty, I feel like talking about the functionality of the opening in terms of the history of the square sleeve would be more interesting of a conversation than saying it was for ventilation. So I'm curious; why is the ventilation answer the most popular? Is it that the movement functionality is just taken for granted? Maybe it is just not that well known? I've also read on forums and blog posts that it is for the ohashori but clearly the opening was being used well before the ohashori was.
There are some characteristics of the modern kimono that are a result of more tradition than function. The ohashori, for example, has become a permanent part of the kimono aesthetic even though it's not really necessary. Maybe the miyatsuguchi has been regarded as something like that.
But this leaves a different mystery. The kanji of the word miyatsuguchi 身八つ口 translates to "body eight hole". Why the word eight?
(*) - Added note:
I came across a garment with miyatsuguchi that doesn't have kakusode (square sleeves) but tsustsusode 筒袖 (tube sleeves). Therefore, the opening is required for garments wrapped at the waist, regardless of sleeve shape. (Image from here.)