NEW YORK—Public high schools in New York City will welcome students back for in-person instruction on March 22, Mayor Bill de Blasio announced on Monday, the latest step by the United States’ largest school system to open classrooms shuttered due to COVID-19. “We have all the pieces we need to bring high school back and bring it back strong, and, of course, bring it back safely,” de Blasio told a news conference. “We are bringing our schools back fully in September, period.” The mayor has made restarting in-person instruction a priority, even as health experts have alleged that teenagers are more likely to spread the virus than younger children, making the reopening of public high schools seemingly riskier than lower grades. Students in Los Angeles, Chicago, and other big cities have been shut out of the classroom since last year, as officials struggle to come to agreements with teacher unions …