You were told wrong. "at the time" is almost always used in a sentence with a past tense. "back then" refers to an entire period of time in the past, as you say, so it cannot be used as in your second sentence.
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In the number 1 sentence, the use of 'at the time' is past tense which is definitely correct. We can use 'at the time' for past tense and for future tense as well.