I think that's so, but that relationship between theory and application exists for every technical discipline. Computer scientists vs software engineers, for example. The line is usually blurry; I tend to operate closer to the applied side of software, for instance, but I still think about and am informed by the theoretical side, just as theory is shaped by the experimental results of application.

"Mathematics" is an odd case because what people call "pure mathematics" is upstream of even the theoretical side of technical specialties. Like, what a theoretical mathematician might call an applied mathematician, I might call a data scientist, because they're closer to pure theory than I am, but still closer to technical application than the theoretical mathematician. It's a super-theory that underpins other theoretical domains.

assigning error bars is purely subjective

I don't know where you got this idea from but it is incorrect. Error bars are used to indicate uncertainty in measurements and they are used to indicate confidence (or lack thereof) in those measurements. Measurement is hard, and precise measurement is harder, so engineers of all stripes use error bars to indicate how precisely their data have been recorded. It's not just a stats thing.

also this:

results change based on the lens through which you interpret the data

happens in every field, including pure mathematics. Look up the axiom of choice if you would like a lot of further reading about the implications of interpreting mathematics through that particular lens. Much as we may long for a "purely objective" language of the universe, free from the limitations of human interpretation, we haven't discovered it yet. The best we can do for now is try to make good assumptions and build from there.

Assigning probabilities is arguably pure math

Not even that is pure math. It depends on your prior knowledge, for example if you think one event is more likely. On the other hand if you don't include prior knowledge/assumptions like one event being more likely you're implying that the prior knowledge behaves in a way that makes your combination of probabilities and data the way it is (for example a flat prior, aka every event has the same likelihood, but in some cases it gets even weirder and would effectively force an absurd prior, so you typically just avoid that by defining some prior knowledge beforehand).

kind of, but you're looking at the wrong end. we're talking like 1600 a month for a policy instead of 12000 (some reasonable numbers from 15 years ago i'm pulling out my ass). 20k a year instead of 150k, just for your practice's liability and malpractice insurance, all because you have the patience to stick your hands in people's mouths (which most medical practitioners don't).

also i don't know if dental drills have changed, but wear ear pro when you use one if you're a dentist, DA or hygienist. the brief time a patient hears it won't damage their hearing (much if at all), but all day every day? My dentist friend is profoundly deaf in the ear that faces toward patients thanks to his drill. that's a risk to one's self most medical practitioners aren't willing to assume.

i tell anyone who wants to go into medicine, doesn't know the field they want, and wants to make a lot of money to shadow a dentist for a week. tell them you want to go into dentistry and they'll get excited (they might be able to sell their practice to you and retire once you're out of dental school). at least that was my experience. I don't like putting my hands in other people's mouths.