Well, since they did accept that "Voldemort tried to cast AK on a child, boom big explosion", and they do suspect that Harry's scar is indeed connected to Voldemort, I don't see any reason for them to distrust Harry's account of the events, especially since they mostly match what they found there.

As for keeping secrets, not speaking about "the sense of doom" and the Azkaban issue to the Order did harm, but keeping secret about partial transfiguration and his ability to transfigure carbon nanotubes allowed him to save the day, so I wouldn't say "keeping secrets has NOT worked well" in general.

If I were Harry, I would probably wait a bit until the emotional chaos (both inside Harry and inside McGonagall) lowers a bit before telling them "hrm, you know, I'm the one who killed the Death Eaters". If I were to even speak about it at all.