Informal writing rarely features narration to any significant degree, which is where the so-called American Style is most "logical" to the extent that we're even really talking about logic. In forums and email and texting, quotes are usually either blocked-off text, or the quotes are used to specifically emphasize an exact character string (often a single word).

Wikipedia is also generally not about narration, and quotes are usually meant to be exact.

With narration, the goal is not to convey exactness rather to tell a story. Interrupting a character's quote to insert ", he said," influences the original meaning (if there even is such a thing) no matter where the punctuation lies. But that's not important because specifying precisely what a character said usually isn't the point of a story.

Furthermore, if you write your own sentence, and finish with quote of an entire sentence, why isn't there a period for both sentences? Brian said, "let's go.".

Looked at this way, it's easy to see why, given the choice, narrators would choose the more aesthetically pleasing placement inside the quotation marks.