I think the organization began its free-fall when Kenny Williams was kicked upstairs and replaced as GM by Rick Hahn.
He had his detractors, but KW was arguably the greatest GM in Sox history, winning a WS and three AL Central titles during his run as GM from 2001 to 2012. They finished second in five other seasons and at .500 or better nine times under Kenny, putting together a 1,014-931 record during his stint as GM.
The KW teams were great fun to watch, even when flawed. And that showed by attendance, which averaged at around 2.5 million between 2005 and 2011.
KW was an excellent evaluator of major league talent, and was able to acquire undervalued talent who made a difference -- A.J. Pierzynski, Jermaine Dye, Freddy Garcia, Carlos Quentin, Juan Uribe and dozens more.
Imagine if he had been able to swing the deal with the Marlins for Miguel Cabrera for (reportedly) Josh Fields, Gio Gonzales and (maybe) Jon Garland back in 2007! KW had that deal sewed up - the only reason that didn't happen is because JR didn't want to take on Dontrelle Willis's salary, which the Marlins insisted on. The Tigers did accept that salary, and the rest is history.
But the bottom dropped out under Hahn, who had only one really good year during his tenure -- 2021. (I don't count the Covid year). Despite having higher annual budgets than KW, Hahn's teams underperformed year after year. He followed up KW's stellar run with a 716-846 record, a barren farm system and consistently dwindling attendance.
KW was much more effective than the overrated Roland Hemond as GM. Only Ron Schueler and the late 40s/50s/early 60s GMs (Frank Lane, Chuck Comiskey, Hank Greenberg, Ed Short) could approach Kenny's record. (Larry Himes built the great early 90s teams, but was forced out before he could enjoy their success).