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I'm responding both to this post and the one above it with this quote, so bear with me. The three games you mentioned, Battlefield, CoD, and TF2, all have very different sorts of teamwork. In CoD, teamwork is really unnecessary, and is in fact discouraged by the mechanics; grouping together is a bad thing when K/D is the main indicator of victory, and a single grenade can take out the entire team. None of the classes are a specific counter to much of anything; people generally make the best class they can with the weapons available, and try to get as many kills as possible.
In the Battlefield games, at least Battlefield 2, teamwork is absolutely necessary, but the form it takes is squads of six people, five of whom answer to the squad leader, and the sixth of whom answers to the commander. Orders trickle down from the commander, and the squads scramble to capture flags. A good squad leader can get his squad in and out of the spawn points, capturing them quickly and with minimal losses. The different classes can be used to counter specific things -- for example, every squad needs at least a couple player of the Anti Tank class, as well as a medic or two -- but they aren't quite as hard of a counter as the ones in TF2. The real power in the game is in the vehicles, and any class can use those.
As for TF2, my other post pretty much explained it. There's only so much any single class can do; a good team not only needs a decent spread of classes, it also needs members who recognize when to switch classes to counter something on the other team, as in the examples I mentioned. Unless both teams are full of rambos, team work is an absolute must; if one team is working togther, and the other isn't, the team that doesn't work together will be steamrolled. The same can be said for Battlefield, but not so much CoD.
As for the Overtime thing you mentioned, Overtime kicks in when the objective is partially captured by the team that would otherwise lose. For example, in a CP map, if the blue team has part of the final point captured, the game will go into overtime until they either cap it, or the meter runs out, and it goes completely back to red/uncapped. In a payload map, overtime lasts as long as there is someone from the attacking team on the cart. It's kind of an acknowledgment by the game that sometimes the the defending team has lost, even if the timer says they won.
To answer the last question, why the strategy in other games isn't as obvious as it is in TF2, I'm not sure. TF2 is almost RTS like in the way each unit counters another. In most FPS games, the differences in the overall abilities of the different classes -- if there even are different classes -- are minimal beyond what weapons they happen to be carrying. I can think of very few games that have classes as specific as the ones in TF2, and most of those are F2P games that came out after TF2.