"ONLY PLAY YOUR TABLE" is a given. What other table are you going to play?
As I see it the OP's question isn't about play so much as it's about strategy.
While I know you can make a final table and even win by only looking at your table and completely ignoring all the other information at hand, I'm not going to pretend like it's not a perfectly good idea to pay attention to things like how many players are left and how big the top say 15 stacks are at that point.
Most of the entrants only care about that cash bubble. Those are all absolute beginners and losers who are willing to settle for one measly little crumb when there are 9 nice big slices of pie waiting at the final table. Each slice is bigger than the one before it, and I just gotta get the biggest slice I can snag for myself. I did not just spend 2-4+ hours(online, but for live I reckon it's as much as 4-5 days at this point), playing my heart out just to walk away with a sad little crumb. I want my slice dangit. If I gotta slit your mama's throat(figuratively of course, as I'm actually pretty wussy in the physical sense) to get my slice, then consider mama's throat slit.
What this boils down to is my having to enter as much as 100 or so times to make the final table. When I do make the final table(note the lack of "if" it's all when baby), I'll snag at least 10 times what the pitiful crumb snatchers take in total, when they settle for their crumbs.
Don't get me wrong. Crumbs are inevitable. I have to accept them from time to time. But crumbs don't fill me up. They don't leave me satisfied. And they sure as hell don't make it so that I can either A. never have to play above my br again, or B. move up in stakes. I'll take the crumbs, I'll take the "losses"(they're all losses till you take down 1st imho), but I want, no I need the slice.
Great I just got myself all fired up and I'm not even planning to enter any tournaments at the moment.
Anyway, in the quest for my slice I want to see just how big of a lead I need, and how much I have to scrap to really have a chance for the biggest slice. That's not wrong, it's right. Wrong is getting blinded by the stacks at your own table, stacks which can give you a false sense of confidence.
Understand, when it's down to three or four tables, you can be chip leader at one table and completely dwarfed by the short stack of another. It happens. So yeah, take a peek, formulate a plan which you can only use at your table. That plan should include taking as many chips as you can wring out of your current table to the next. But don't assume that having a healthy to big stack at one table guarantees being above average. There are just too many factors that are completely out of your hands.
Factors like, the cards you're dealt(card dead at a heavily aggro table's pure agony and chips away at your stack like crazy), table dynamic(you can't always shift this dynamic with ease, and you don't always get a chance to play with or against it properly), and yes luck. Luck's a funny thing, akin to fate, and it can turn a big stack into a short stack in no time.
Pray for the rush, or as the French say on those saison II episodes of WPT, "Le Super Rush".
Anyway, check, or don't check. Both are correct. Still there are very valid reasons to check late in a tournament.