Mating Tactic!

This is from round 5 of 2017 PanAm InterCollegiate Chess Championship

White to move and mate in 9

Robson vs Zherebukh

what about:
1) R x g7, R xg7
2) Rd8+, Qg8
3) R x g8, K xg8
4) And then the white Queen can mop up the pawns.

Vipinzamvar, why does one need to take pawns if one can checkmate?

is it possible with h4

1) h4, a5

Rg4, b5; Qe5, Qh6: h5, Qc1+; Kh2, Qc2+; kh3, Qd3+; Rxd3; e4xd3; h6, Kh7; Rxg7, Kh8,; Rxg8, kxg8, Qg7+ mate