In my latest reading of volume 4 of John Meier's A Marginal Jew: Rethinking the Historical Jesus, Meier talks about the co-existence of strictness and leniency in Jesus' approach to the Torah.
According to Meier, Jesus was stricter than the Judaisms of his day and thereafter on the issues of divorce and oaths. Jesus prohibited divorce and remarriage, which set him apart from Second Temple and (later) rabbinic Judaism. Whereas the Dead Sea Scrolls are concerned about people having more than one spouse, they do not contain a blanket prohibition of divorce. And, while scholars have compared Jesus' position on divorce in Matthew 19:9 (which has the exception clause for fornication) to the position of Shammai, Meier believes that Jesus and Shammai had different views on the subject. Shammai believed that a man could divorce a woman if she shamed him, whereas Jesus prohibited divorce, period.
For Meier, Jesus was also revolutionary on the issue of oaths because he prohibited them altogether. Even Jews who had issues with oaths, such as Philo, still believed that they were permissible on some level because they were commanded in the Torah. Consequently, there was discontinuity between Jesus and Judaism, and Meier views that as a reason that Jesus historically held these positions on divorce and oaths.
But Jesus was lenient on the issue of the Sabbath. Meier does not regard as historical the interactions in which Jesus gets in trouble for healing on the Sabbath, for rabbinic Judaism largely did not prohibit that. Moreover, Meier does not believe that Jesus getting in trouble in Mark 2 for his disciples plucking grain on the Sabbath is plausible. While Meier acknowledges that there were voices in Second Temple Judaism (i.e., Philo) that considered plucking grain to be work, he does not think that the Pharisees would have popped out of nowhere near the grain-fields, accosting Jesus and his disciples for plucking grain on the Sabbath. Moreover, according to Meier, had Jesus historically made the Scriptural bloopers he makes in Mark 2 (i.e., referring to Abiathar, when the priest in I Samuel 21 was Ahimelech), the Pharisees would have laughed at Jesus rather than taking him seriously and getting angry at him.
But Meier regards Jesus' statements that people on the Sabbath could pull an ox or a man out of a ditch to be historical, even though they were later attached to "Sabbath controversies" contrived by the early Christians. For Meier, Jesus had a common-sense halakah when it came to the Sabbath, one that sympathized with peasants who felt they needed to pull their ox or a person out of a ditch. Meier speculates that Jesus was responding to the Essenes or stricter voices within Pharisaism.
Why Jesus was strict in some areas and lenient in others, I do not entirely know. One point that Meier makes is that creation played a significant role in elements of Jesus' halakah. Jesus opposed divorce and believed that the Sabbath was less important than human well-being on the basis of the creation stories in Genesis, for example.