The reason you're hearing the written sentence in your mind's ear with a not in it
that way, the sentence is grammatical.
At all is a Negative Polarity Item, which means that it can't appear grammatically at all,
unless it's inside the scope of a negative trigger; and by far the simplest and most common
negative trigger (linguists would say the "default" or "unmarked" trigger) in English is not.
Without, which is a negative trigger, appears at the beginning of the sentence and doesn't command (and therefore can't trigger) at all, which is in a subordinate clause.
Thus, your brain obligingly supplied not for your mind's ear, because your brain knows English syntax, whether you do or not.