Most interactions in the world are of the form "I have an idea of what will happen, so I do X, and later I get some evidence about how correct I was".

Perhaps, but a the doctor in the OP did not just happen to later get some evidence about how correct he/she was; instead, after formulating a hypothesis, the doctor ran a test specifically to test the hypothesis. That is practically a textbook example (albeit a fairly short/simple one) of the scientific method at work.

though I endorse promoting reflection on the fact that this is what is happening

And that was really my point. It is worth noting that the scientific method is really just a very rigorous formalization of common sense reasoning. I think that demystifying science among the non scientifically sophisticated population is actually a step in the direction in which the OP gestures.

Normal interactions with doctors are low on scientificism in this sense, though they are heavily utilizing the output of previous scientificism to generate a judgement.

This also is true; even if one can't expect the full-on House M.D. treatment each time one goes in with a sinus infection or strep throat, many of the protocols that the doctor follows and the medicines that he/she prescribes were developed/tested with a high degree of scientific rigor.