Apparently, what this one guy meant is Merkurjev's proof of Merkurjev-Suslin theorem, which was writen down by A. R. Wadsworth and later by W. van der Kallen. The latter paper starts with the statement of Matsumoto theorem and the proof does indeed use cohomology of sheaves of K-groups on Severi-Brauer varieties.
As for the nicer proofs of Matsumoto theorem, one can take a look at A new approach to Matsumoto's theorem by K. Hutchinson or $(t^2−t)$-reciprocities on the affine line andA new approach to Matsumoto's theorem by K. Hutchinson or $(t^2−t)$-reciprocities on the affine line and Matsumoto's theorem by F. Keune.
