I don't think guys should be pushed, that's not very nice and violence is never the answer.
Ok,but, I think it's more of a function of guys showing you they're ready. The Sox never pushed Mookie, he just started mashing and they really had no choice but to push him. Guys will tell you when they're ready and it's not all stats, there's other things they're looking at too but if a guy is putting up video game stats for a long duration it's hard to imagine what else they could ever learn at a level.
I don't think Yorke gets to the majors before Downs. He's in low, not high A. He still has to get to Greenville, Portland, and Worcester before reaching Boston while Downs just has to get up to the next level. Even if Downs is a bust I could still see him getting the majors sooner if a guy gets injured or they need the depth option on a bench role. Downs will likely be added to the 40 man by year's end so, you probably see him in some capacity next season. York, if he mashes, I don't think you see him in 2022. The earliest we see Yorke is mid-2023 I think. But I could be wrong.