When he walked into the mayor’s office 12 years ago, businessman and billionaire Michael Bloomberg took charge of a damaged city. The World Trade Center was a windblown construction site, barely cleared after the attacks on Sept. 11. Tourists were afraid to come to the city; residents were afraid to stay. The budget was a disaster, $3 billion to $5 billion in the red. In a modest speech at an intentionally modest inauguration, Mr. Bloomberg nevertheless pledged to rebuild and renew New York and to keep it “the capital of the free world.”