US News & World Rankings: Do They Matter for Residencies?

Short answer is no.

Long answer is of course not.

Those rankings look at a ton of factors that the residency programs can't control, or care about from a training stand point. Many of those factors are based on the nurses and patients overall satisfaction with their stay. For country hospitals, those patients are generally not as "friendly", and those hospitals (and residency programs) naturally score lower. If you want the best training, its all about repetition and city. Busy places will teach you the most and you will see how to manage patients. Those programs have good names, but also have good undergrad names and they ride on that.

People say they do not matter, but HSS which 90%+ of people on this forum would rank #1 is also #1 on the usnews. Not saying all good residency programs rank highly but certainly there is some degree of correlation

These rankings have nothing to do with residency or academia. If they did, Cedars-Sinai or Beaumont would certainly not be ranked higher than WashU, Penn, UCSF... They are primarily determined by patient outcomes, surveys and to a limited extent, reputation.
These are designed to guide patients, not medical students.