You are forgetting that the 2004 team had high priced free agent pickups such as Manny, Johnny Damon, and Keith Foulke, two veteran starting pitchers that combined to make $29M, and a well-paid home grown player whom the Sox were able to lock up long term in one Nomar. There was a reason that Theo had to go out and pick up all those scrap heap players you mentioned: the Sox farm system was dry at the upper levels, and they needed guys to round out the roster, which had become very top-heavy in 2002. Necessity is the mother of invention.
The 2002 team, although it won 93 games, had a number of holes to fill: 1B, 2B, 3B, DH, and bullpen and starting pitching. Their payroll had Manny ($20M), Pedro ($15.5M), Nomar ($11M), and Damon ($7.5M). The owners had just bought the team and the ballpark, a facility which needed some significant upgrades in order to expand their in-game revenue. Local revenues were strong, but not on as strong as a footing as they are today. Plus the FA market going into 2003 wasn't exactly stellar. Basically, a lot of holes, and not much cash. And also not a lot of great FA's out there that off-season; even the Yankees were relatively quiet. But the Sox did find, as noted, a bunch of unwanted guys that they were able to use to fill those holes, and then some. But they also got lucky, in that most of them suddenly started having career years.
After the 2003 season, they shored up their biggest areas of need by trading for Schilling (and his $12M contract) and signing Foulke; and this was after trying to get A-Rod. The 2004 team had the largest payroll ever for a World Series Champion. Poor teams don't add such major contracts to their payroll.