I found this post late after the NYT changed the title and the HN title is "didn't use to", so was confused on what the problem was because I figured the post you replied to simply misspelled "didn't used to".
Anyway, was curious myself and found this [0] which looking through comments mentions what I thought the problem was, spelling. Others say it's intended. But then this one [1] has far more discussion on the topic and a comment under the question (from `a retired English grammarian`) "How you want to spell /'dɪdən yustə/ is what causes all the problems. "[1].
"didn't used to" simply doesn't look right to me (I can't recall ever having come across it), but the /'dɪdən yustə/ explanation is the simplest way to make sense of correctness.
Another interesting discussion[2] (this seems to be a popular topic) has a few ESL teachers that say they teach "didn't use to" and taught "didn't used to" as being wrong and were surprised to find that "didn't used to" is quite common with usage being correct depending on which rules a person chooses to acknowledge. There are apparently rules that make "didn't used to" correct and they are talked about in the others, but more so in this one.
[0] https://english.stackexchange.com/questions/385006/didnt-use...
[1] https://english.stackexchange.com/questions/8816/whats-the-n...
[2] https://languagelog.ldc.upenn.edu/nll/?p=2756
