I ran across the phrase "did use to" in Bunyan's Pilgrim's Progress, in the sense of a past action: "[This giant] did use to spoil young Pilgrims with sophistry." It looked odd - I would have expected used instead of use, and today we would dispense with "did" in that context altogether. I suspected that the -d got added in for either of two reasons. It was either a sound-mistake based on use being followed by to, which just fits, if you say it aloud to yourself a few times; or, someone thought it just had to be past tense "used" to go with the past tense "did," which would be foolish, but exactly the sort of thing that people who like to make up extra, unnecessary rules about English usage are prone to.
Used to is later, showing up first in the 19th C, and pushing out use to entirely by the 20th. Before that, it is all use to, meaning "habituated" going back to 1400. There doesn't seem to be any reason for it, it just happens, suggesting that the sound-mistake is the explanation. If one goes over to grammar and usage sites, however, one can find people who not only insist that used to is correct because of the match in tenses, but asserting it forcefully and complainingly, as if only fools and destroyers of Proper English would ever think otherwise. This type of error is common among those folks. Some usage that came into the language rather late - say, 100-200 years ago - which was unknown to any speaker of English, even the most educated and obsessively precise before then, is regarded as correct, on the basis of being popular for the fifty years before the birth of the speaker's Aunt Agatha, who was an expert on such things.
If you contradict them in this, they accuse you of encouraging all manner of license, having no standards, and contributing to the general decline of culture. Some people are just naturally Great Deplorers of Things.
BTW, the "did" in that phrase comes from another great destruction of all that is good, when the invading Anglo-Saxon speakers married a fair number of Brythonic women in the British Isles. Welsh and Cornish have, and had, that "did" as part of their language, and it just refused to leave, conquerors or no, when those Danes and North Germans showed up. "What did you eat?" would have only been "You ate what?" in the Germanic languages, and still is. So the did in that phrasing, is just wrong, wrong, wrong, and any decent speaker of Anglish would know that.