For comparison, the top-rated English language feature films of 1986 on IMDB are:
Aliens (8.5)
Platoon (8.2)
Stand By Me (8.2)
Ferris Bueller's Day Off (7.9)
Hannah and Her Sisters (7.9)
Parting Glances (7.9)
Blue Velvet (7.8)
Down by Law (7.8)
The Name of the Rose (7.6)
Hoosiers (7.5)
A Room with a View (7.5)
Salvador (7.5)
The Fly (7.4)
The Mission (7.4)
Children of a Lesser God scores a 7.0.
Of the 50 films nominated for Best Picture in the 1980s there are 7 that I haven't seen and two of them are from 1986: Hannah and Her Sisters and A Room with a View. Out of the three nominated films that I have seen it's a close call between The Mission and Platoon for my vote as to which film should have won. I'll throw my vote to The Mission.
The score for The Mission is one of the greatest ever written. Ennio Morricone was nominated for it, but didn't win (in fact he's never won an Oscar in competition, having to make do with an honorary award). I haven't seen the film that won for Best Score that year ('Round Midnight), but it would have to be one hell of a score to deserve beating The Mission. In all likelihood the Academy simply got it wrong, but of course I'll have to see 'Round Midnight to make that judgment.