This site presents a viable theory that Mallory and Irvine summited Mount Everest on June 8, 1924. They climbed a moderately difficult route just above the Couloir (pictured above) which had been identified by E.F. Norton two days earlier. They did not climb the Second Step, but instead took a lower route to bypass the obstacle, just the way Mallory described in the 1921 reconnaissance report. Although Mallory climbing the ridge route is a popular theory, Mallory stated the ridge route was impossible.
Mallory described his potential routes to John Noel, the 1924 expedition photographer, in the days prior to his climb. Mallory stated his first choice was to go up the Couloir to access the north-east section of the final pyramid. His backup route was to cross above the top of the Couloir and access the ridge of the East Face at a point above the Couloir. This second route matches with the sighting of Odell at 12:50.
The myth they climbed either the First or Second Step was a deliberate deception created by E.F. Norton. Odell sent a mountain dispatch down to Norton at Advanced Base Camp stating they were seen at the Final Step (Third) at 12:50. Norton destroys the dispatch and then falsifies the altitude and time in a report to The Times.
When Mallory was found in 1999, the multiple searches of his body and having to bury him, dig him up, and then search again ruined the ability to determine whether they made the summit. Had Mallory made it to the summit, he would have collected summit rocks and put them in his pockets. In the haste to locate a camera, they destroyed the evidence that would have shown whether they made it to the top.
UPDATE: I have some comments on the 2019 search which is described in National Geographic. I had nothing to do with that search.