Any time you hear of someone saying, "I sent a letter to X school and then got an interview from X school, these things definitely work," immediately stop paying attention to this person. It is conceivable that many other things this person says will also be layered with plenty of retrospective bias and unfounded assumptions. No one will ever know if it was the letter that got them an interview or because the committee was going to interview them anyway. In a high-stress situation such as the med school application process, you will hear of many similar stories that should get you a good chuckle and absolutely nothing else. I'm not saying that people are dumb, but people in general are very susceptible to generating false narratives to fit the empirical data they see, especially in the form of overly simplistic stories as they grapple with an overly complicated process.
For Yale specifically, Richard has always been very transparent in that he reads LOIs post-decisions and takes them seriously. He mentions this outright in many WL acceptance phone calls (including my own). There is no evidence anywhere else that sending a letter to admissions at this stage gives you any leg up on getting an interview. However, basic human logic might suggest to us that if any student applying to a school can do the same thing in a blink of an eye (i.e., simply write a letter pledging allegiance to a school they have not seen in person or otherwise interacted with
😏), the effect of such a letter is probably not fantastic enough to be a worthwhile recommendation.
For ITA emails, always send them. The worst a school can do is ignore you, the average a school a do is decide they weren't going to interview you anyway, the best a school can do is save you hundreds of dollars. Win-neutral-win. There are certain schools known to be very inflexible with such special requests (ahem, Georgetown
@Jalby) and will tell you as such once you write the email. 0% harm done in all cases.