I don't see why we're excited to GII here with TPGK. You can only get 1 or two streets of value with this hand. I'd check-call, check-call, check-fold. Or check, bet, bet if he checks behind OTF. You'll be surprised by how some people call you down lighter if you check a street, especially since there is a FD out there. Also, our hand cannot stand any heat if we get raised on the flop. AK beats us, we're chopping with KQ that are doing dumb min-raises, crushed by sets, barely ahead of the nut flush draw. Calling or shoving turns against that range is spewing money. By checking we can also induce some bluffs or get some low/mid PP that think they're good and put out a "protection bet" lol. It's also good to have semi-strong hands here in our checking range. People generally bet here with all their value hands/air/Kx, so checking narrows our range to QQ/JJ/marginal hands most of the time, and we can get barreled off by any decent player.
I'm fine with 3-bet pre. We retake the initative, can bluff on decent flop textures, or take it down pre. Decent blockers to premium hands and their continuing range helps a bit too.