Here's why Psych stopped using flashbacks in season 6. For the first several seasons, flashbacks played an important role in the USA series; typically, an episode would open with a scene set during Shawn's childhood. These opening flashbacks showed young Shawn (Liam James) learning a lesson from his father, Henry (Corbin Bernsen), and this would normally relate to whatever problem the adult Shawn (James Roday) would experience in the main story.
Flashbacks in Psych mostly focused on Shawn interacting with Henry, but young Gus (Carlos McCullers II) was often incorporated into them as well. This concept became a formula for the show that was consistently used from the Psych pilot episode all the way through season 4. They were used in the first few episodes of season 5 as well but gradually became less and less frequent. In Psych season 6, only a few episodes featured flashbacks at all. At this point, the formula was gone and the idea was dropped completely after the season 6 finale.
Shawn and Gus' friendship was the central relationship, there was also the father-and-son dynamic of Shawn and Henry. At the beginning of the series, Shawn had a lot of issues with his father, including the fact that he blamed him for his mother's absence. Shawn felt that the two of them couldn't get along, even though Shawn greatly depended on Henry's advice when it came to solving cases. By the time the show was in its fifth and six seasons, their relationship had been properly fleshed out. So during this time, there wasn't much of a need for their history to be explored further. The problems between them had been mostly resolved.
Furthermore, the age of the actors had become a concern in the later seasons. The child actors who had played young Shawn and young Gus were in their early teens. Psych had already been in a position where it needed to recast young Shawn; it was when Skyler Gisondo replaced Liam James that the show's flashback formula started to disappear, and the scenes with the new actor were never received as well as the ones with Liam James, who successfully captured the mannerisms and personality of the adult Shawn. The loss of Liam James - and the idea that they would soon have to replace Carlos McCullers II as well - could have been an important contributing factor.
Also, there was only so many lessons that Shawn could learn from his father that the series could still explore, so it did make sense that Psych would have to move on from flashbacks eventually. For some, this was disappointing, as they created memorable moments, particularly in episodes where they factored into the main story. One example of this happened in season 2, when Shawn spent an episode trying to figure out which kid in his classroom fired a spitball at the teacher, only to discover at the end of the show that it was Gus all along. This scene allowed the young and adult versions of Shawn and Gus to actually interact in Shawn's head. For reasons like this, the flashbacks remain a beloved part of Psych's early seasons.