Personally, it makes sense for "scare quotes" to not contain punctuation, as they are not complete sentences. But it doesn't make sense for direct quotes to not contain punctuation, as in `He said, "Hello there".' He didn't say "Hello there", he said "Hello there."
(It might seem logical to have two periods in that case, but it's ugly, so the second one can just be omitted for maximum conciseness. That's why the period goes inside the quotation marks. Similarly, it would be confusing and ugly to pretend to end a sentence in the middle of a sentence, so quotes that are not at the end of the sentence "end" with a comma. The period is a pretty strong message to pause, and you don't want to overuse it.
IMO.)