  Lambrechts, Turchin and Voli\'c proved the Bousfield-Kan type rational
homology spectral sequence associated to the $d$-th Kontsevich operad collapses
at $E^2$-page if $d\geq 4$. The key of their proof is formality of the operad.
In this paper, we simplify their proof using a model category of operads. As
byproducts we obtain two new consequences. One is collapse of the spectral
sequence in the case of $d=3$ (and the coefficients being rational numbers).
The other says there is no non-trivial extension for the Gerstenhaber algebra
structure on the spectral sequence.
