It is not certain that Jesus used the LXX. At an early age he was apparently focussed on the centre of Hebrew learning in Jerusalem, and no translation of Hebrew or its Semitic family could fail to be other than a pale imitation of languages that were like no others, particularly Hebrew itself, due to its particular culture. In his contests with the Jewish religious leaders he would have needed to have been closely familiar with the Hebrew texts, and there are strong signs in his detailed references to their Scriptures that he was familiar with them.
It may be simply that the gospel writers naturally used the familiar LXX in quoting him in their
Greek gospels, where it was a fair represention of the original (and it quite often isn't). There is certainly some value to the scholar in the LXX (mostly in checking uncertainties in the Hebrew sources), but its role is much exaggerated by certain sorts of religious people.