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While I was watching an ESPN documentary, I heard the line "The team would go on to win the championship." I have heard some people overuse this 'would' when telling stories. Is there a rule to follow?

My opinion is that I would use this form only if the statement is at a different time than the rest of the story AND I am returning to that base time to continue the story (like a parenthetical aside). However, I've heard it used many times in a conclusion.
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A bit of context, pls? I think I've heard that myself, but ...
Was it something like:
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In 1869 he put the Red Stockings on the road, traveling 11877 miles and drawing
some 200000 spectators. The team would go on to win 91 consecutive games ...
Baseball Anecdotes - Page 7

by Daniel Okrent, Steve Wulf - Sports & Recreation - 1989

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?
Sure, it was something like that. The construction 'would go on to' is also very common. 'Won' conveys the exact same idea, doesn't it? So does 'went on to', I think.
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I think you have a point. Now, in trying to find an explanation to it:
Would, as you know, it's used in a lot of storytelling in order to create the imaginary atmosphere of the situation in the past. And this is not for showing typical or repeated action, as would does in other situations.

I think this particular wouldcreates a final punch, telling us that we re-enter reality from story, history or pastEmotion: smile
I agree. I think the usage that bothers me is when it is used more frequently in a story/biography, rather than sparingly .
Glad we agree on this interpretation.
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I think that this is a case of would used as the past tense of will.

The team would go on to win 91 consecutive games. It is the simple past narrative form of will.

If you were writing in the present tense the sentence would read: In 1869 he puts the Red Stockings on the road, traveling 11877 miles and drawing some 200 spectators. The team will go on to win 91 consecutive games.