The outside sentence can continue after the quote; the terminating period outside of the quote is necessary if the sentence is ending there.

We know that the quote has ended with the closing quote, and so we can agree on the convention that there is an implied period there, if the quoted material is a complete clause with subject and verb. If it needs some other punctuation like a question mark, then that is explicit.

> He asked me "what time is it?".

The ? ends the question, the . ends the sentence that contains the embedded question.

The enclosing sentence can continue:

> He asked me "what time is it?" but hurriedly walked away as I glanced at my watch.

Now you positively, definitely cannot remove the period after "watch". Why would you remove it if the words "but ... watch" are deleted?

This stuff is simply too important to leave in the hands of people who have never written a compiler.

I meant my point mostly tongue in cheek. Redundancy is not necessarily bad.

I do have trouble reading your sentence, though. The change of voice in a declarative sentence to interrogative without a stop, is kind of jarring. In that one, I'd leave off the quote and question mark, I think.

That all said, I think I mainly agree that it is all guidelines and mostly driven by reasons we no longer remember or care about. The rule I remember hearing was that the smaller character of the period was preferred inside the larger character, as the period was prone to break if done in the other way.

> He asked me what time it is, and ....

Now we have an embedded wh-clause, not an embedded quote. Embedded wh-clauses are not questions.

(One piece of evidence: subject and auxiliary are not inverted, appearing as it is rather than is it.)