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Editing tweets is against the core concept of Twitter.

I would consider “stories” against the core concept of twitter as well.

why not have a edit history but still show the most recent tweet at the time?

It doesn't mesh well with the retweet functionality if a user can retroactively change the content.

Then have the retweet keep the "version" of the tweet that they retweeted, maybe with a flag to say "this tweet has since been changed"?

That doesn't seem to be a problem on Facebook. Shared posts can still be edited later.