My mother started by taking my temperature rectally herself then, at an age I have not recorded but that should be around 6, switched to having me insert the thermometer rectally myself.
I however had seen cartoons and comic books where patients got their temperature taken in the mouth. I asked mom why we did not do it this way (it seemed less cumbersome) but she said it was unreliable and I should stick to rectal.
Same in hospital - the nurse handed me the thermometer, I thought it would be like in comic books and slowly moved it to my mouth, she tut-tutted and said "here at the hospital we do not take it in the mouth".
Regarding breaking thermometers, I think we broke a mercury thermometer by dropping it on the floor. Mom complained years later about the difficulty of collecting mercury droplets.