I cannot talk instead of women but I assume that the males were more embarrassed by rectal temps because it was their mother or a female sibling or a nurse who was in charge of inserting in their rectum the thermometer
Now normally there are no more rectal temps but in the past, when I was young, it was the common practice for a child at least until 14 age, it was generally his mother who did it and there was no way to refuse. After 14 age it was depending on families but normally it could be taken by the boy himself without help.
At school for boys only, it was the same, the permanent school nurse used to take the temperature rectally each time it was necessary or even when unnecessary to check that a boy student was not lying about a disease to be exempted of Pe lessons or swimming lessons.
So when you arrived at her office, her first reaction was to instruct you to remove your trousers and pull down until ankles your brief to let her take your temperature rectally, it was of course embarrassing but it was her who had the power to decide how check your health state. She applied this rule even with the older boy's students at last grades of high school.
So at home you could take your temperature alone and sometimes in some families teenagers could take it orally even if everybody was thinking at that time that it was less precise than taken rectally, but not at schools. A high school boy of 17 age had to remove his trousers and pull down his brief like a younger student for a rectal temps when he was in the nurse office.
It was the same in boarding schools and summer camps according to many relations.
I assume that as the boys were considered as less mature than girls of the same age and also because their shyness towards nudity especially when it concerned the butts was not considered as relevant, they were more often and more longer submitted to rectal temps than girls
Anyway It is completely different today