(This may not be answering your question...I just reread it.)
It would depend. If I've fallen behind during the early levels of an MTT, I'm not in a rush to start shoving and hoping to catch-up unless I'm really short on BBs. Lets say its a 1500 starting stack, blinds at 10/20 and I lose half my stack. I still have 37.5BBs left, so I'm not desperate to start shoving and trying to double up just yet. However, I know that when blinds go up again, Ill be chopped down to ~18BBs and I may have to include shoving as a strategy. Not necessarily to just double but to survive the next level of blinds that are coming up.
Early on, I don't see shoving as a necessary strategy because the risk isn't worth the reward. If I shove, I either: A)Double up, but have a long way to go still so even I have more chips now it hasn't affected the outcome of the game significantly. or B) I bust out. Too much risk, not enough reward. However, I'm not saying that I wont shove during the early stages, it all depends. If I'm holding AA in BB and I'm looking at 3-4 bets before me, I'm getting my money. Usually when I am shoving in early stages, it's post flop more and a lot less preflop.
However, as the tournament continues/blinds go up/BBs decrease, then shoving becomes a more necessary tool. For me, If I'm over 50BBs, I'm sitting good. I don't have to worry about PF shoving anytime soon, but once it gets down -50BBs, then its time to start considering. Not saying I'll just start shoving now, but I'm aware that if I am not picking up chips, the amount of BBs I have will decrease very quickly. Lets say I have a stack of 1000 and blinds are at 10/20, I have 50BBs. But as soon as blinds increase to 15/30 (assuming I still have 1000 chip stack), my BB drops to 33. That's ~66% decrease in BBs. At 20/40, my BBs has dropped to 25. Which is why, they say to increase your raising range as blinds go up so you wont be put into this situation. I may not shove a lot sitting on 33BBs, but I might raise more and be more prepared to 3-4 bet shove with that size stack if I need to.