Apparently, what this one guy meant is Merkurjev's proof of Merkurjev-Suslin theorem, which was writen down by [A. R. Wadsworth][1] and later by [W. van der Kallen][2]. 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*][3] by K. Hutchinson or [*$(t^2−t)$-reciprocities on the affine line and Matsumoto's theorem*][4] by F. Keune.


  [1]: http://www.ams.org/books/conm/055.2/
  [2]: https://doi.org/10.1007/BFb0074799
  [3]: https://doi.org/10.1007/BF01436072
  [4]: http://dx.doi.org/10.1007/BF00533156