For Chekov, though, I can forgive him running down the halls yelling "I can do dat!" simply because, well, he did do dat. Even with only 10 min. on screen, he contributed more to a film than his character ever did in previous movies. If he failed to save Kirk and Sulu, or if he failed to calculate the correct course, Earth would have been destroyed. He was as crucial to the mission as the rest of our heroes.
I don't agree with that. Alone his scenes in TWOK are worth far more. In TVH, he becomes the focus of attention for quite a while. In TUC, he leads the investigations aboard the ship.
You listed technicalities that he was involved in, I'm talking about character moments.
I disagree with this, too. We all love TWOK, but really, Captain Terrell basically served the same purpose as Chekov, and became the highest-ranking redshirt in the process (in that, Terrell still would have tricked the Enterprise, betrayed the crew, and reveal Khan's plot -- and just like Chekov, he even resisted in the end. The only difference between Terrell and Chekov is that Chekov lived). On top of that, someone else on the Enterprise was doing Chekov's duties at tactical before he recovered for the climax (which curiously happened before in TMP as well).
In TVH, his mission and capture was really just one played up for laughs (albeit more convincingly than "I can do dat!"). But even then, Uhura brought back the nuclear goods. (and how is this a "characte rmoment?" His capture was a plot device). He was largely played for laughs and in hindsight, didn't really contribute much to the mission aside from being Uhura's sidekick.
And finally, in TUC, while he lead investigations, he was wrong a number of times (the kitchen phaser and the boots come to mind)... Spock and Scotty unveiled more clues (and outrank him), with Spock leading the investigation himself. No one would ever question Chekov's dedication to the ship (I certainly won't), but again, he was played up both for chuckles and exposition.
But I don't really think of, say, his saving Kirk and Sulu as a technical moment. He wasn't in a position to save them. He actually had to get into that position when the crisis came. He stepped up to bat and delivered. And when the crew was coming up with their final gambit, they turned to Chekov to contribute and not an older veteran from the same department.
Chekov isn't exactly cut from Shakespearean cloth in XI, but he did a heck of a lot more than be a victim or comic fodder or an obligatory character.