If you look at this question from a strictly medical point of view……
Let's see, tell patient to sit on exam table. If they are in an exam gown then you need to have then lay on there side or on their tummy. Get thermometer, shake it down, get the vaseline, apply lube, insert thermometer, wait 2 minutes and then pull the thermometer out of the butt hole and tell to just sit there and wait. Wow a lot of work and time to get someone's temperature. If not then stand and pull pants and underwear down and bend over? Or lay on table and pull pants down or lift skirt till you can expose the anal hole and prepare to invade.
The evolution of the thermometer, at least in modern times, was to make it easier and faster to obtain. I am sure everyone has seen doctor's offices and how the exam rooms are lined up and the priority is to have patients in every exam room and ready to be seen by the doctor.
Ie………. It is all about the all mighty doctor and the dollar. I have worked in the doctor's offices and I can tell you part of the job is keeping up with the doctor and making sure he/she has a patient ready to see and do not make them wait. I have also seen this with surgeons waiting for their surgical patient to be preptped and so on and he was pissed he had to wait an extra 20 minutes because the patient was extremely emotional, scared, nervous and worried about her family and needed extra time for some TLC before we proceeded with prepping her.
Anyway, in the actual medical setting I do not think you would even be able to find an old glass mercury thermometer. In the hospital they had the temp on the wall and that had both oral and rectal.
The old fashion doctor who believed rectal temps were the best was to get the most accurate temp probably have all passed away. I do have a few rectal glass thermometers and do use them on those that need a temperature taken.
But, when time allows it is best to expose the ladies buttocks, spread her cheeks, apply a generous amount of vaseline to that exposed anal hole and insert the thermometer into her rectum.