I did something very similar as a project when I taught AP statistics a few years ago. It was relatively effective and my students left that class with that project as one of the ones they felt was the most impactful, since it showed them how statistics was actually used in real life.
I believe that the most recent topics for this project were linear regression and experimental design, and I generally asked my students to summarize the methods and results. My goal was to have papers that did not take a large amount of technical knowledge to understand what, to some extent, the study was measuring and how the experiments or observations were carried out. I wasn't fully satisfied with the papers that I found, but some that served the purposes I wanted them to. It should be noted that I taught at a "medical field focus high school." The papers I ended up using included:
Is grip strength a predictor for total muscle strength in healthy children, adolescents, and young adults?
The Prevalence of Dental Anxiety in Children from Low-income Families and its Relationship to Personality Traits
Validity of self-reported height and weight in a Japanese workplace population.
However, I'm not sure if you'd consider this subject matter "important and useful." One other possible paper that you could use snippets of that is well-known, relatively accessible, and has straightforward statistics :
Cognitive and Attentional Mechanisms in Delay of Gratification
which I don't think is "the" Stanford marshmallow experiment of lore, but could be potentially more interesting to students interested in psychology. In general, older psychological research seems to be the most accessible to students without domain specific knowledge.