My opinion is, since the DTI holds Kirk in such low regard because he's their worst offender, wouldn't they dislike Spock just as much because he was always by Kirk's side during those offenses?
Not necessarily. Kirk was disliked by the DTI because of what they perceived to be his consistent disregard for temporal regulations, as well as his reckless approach to time travel (going out of his way to interact with people from 1986 in order to steal two whales).
Spock aided Kirk in his endeavors, but he always seemed to have a much greater awareness of how to approach time travel, something that the DTI would appreciate. Their branch of the Federation exists to prevent tampering with the timeline. It stands to reason that they would need people who understood temporal mechanics in order to carry out this objective.