I suspect that the focus on "screen time" originally came about because the genre of low-effort studies that try to squeeze headline-grabbing correlations out of "time spent watching TV" needed to switch to a broader measure because conventional TV has dramatically waned in popularity over the past ~15 years.

As far as I know, the large-scale freakout happened after the American Academy of Pediatrics released a report recommending limits for "screen time" specifically for young children, which grabbed a lot of headlines. They later walked it back and basically changed the recommendation to be against unsupervised media consumption, and that grabbed a lot fewer headlines.