Mayor Bill de Blasio’s eyes watered and his face flushed. His voice faltered and, at one point as he struggled for the next sentence, his two college-aged children each placed a hand on his back to steady him. The mayor was outlining the city’s wide-ranging, $850 million initiative to support and treat New Yorkers suffering from mental illness, and he grew emotional as he framed the effort to help with his own family’s struggles with depression.