WEST PALM BEACH, Fla.—A 30-foot Houston Astros logo sits just outside the ballpark here, buttressed by an oversized orange and blue banner that reads, “2017 WORLD CHAMPIONS.” The team earned that designation by beating the Los Angeles Dodgers in seven games in that year’s World Series.

This monument to their victory served as the backdrop Thursday morning, when the Astros finally said “sorry” for the sign-stealing scandal that has tainted the title in the eyes of many and thrown the franchise into turmoil. But Houston owner...