But seriously, how difficult is delivering a shot?

Source: my mother, recently retired medical doctor. You guys got a "Hello" from her by the way. Feel free to ask follow-up questions, just keep in mind standard disclaimer that this is not in any way a medical advice, just fact checking.

Intramuscular injections that are recommended for adrenaline/epinephrine in case of shock are up there among the easiest ways for untrained person to break needle. It's not common, but happens mainly due to lack of experience. Neither of us knows precisely what are the competences of 'basic EMT' in USA, but through know-how on IM injections is a must. Panicked person who probably does it for the first time in a life-threatening situation is much more likely to mess up than fairly used to it nurse or EMT.

Main problem with needle breakage is the fact that it can be brushed off as pronounced after-effects of condition that required fast administration of medications (seizure, anaphylactic shock). Muscle pain, general discomfort, muscle spasms or cramps (EDIT: and in case of anaphylaxis it can even potentially not show a typical immune response for foreign object) that often occurs after such situation can 'hide' the pain caused by needle tip in some cases. We don't know how it looks in USA, but in Polish hospitals it's among reasons why many routine procedures are done in pairs. For example one nurse preps the patient, the other prepares and makes an injection and in the end you have two sets of eyes capable of catching errors. In this case: procedural errors and results like the mentioned needle breakage.