I took a look at the 1946 World Series box scores just to see if it was possible to determine which aspect had a bigger impact on Williams' performance in that World Series, the wrist injury or the shift. I mentioned above the lack of extra base hits, on which the shift would have little impact. Also, during the regular season, Williams struck out 44 times in 672 plate appearances (once per 15.3), while drawing an AL leading 156 walks (once per 4.3 appearances) [Ed: Yes, he truly was an amazing hitter, especially when you consider he hit only 6 fewer home runs than strikeouts]. During the Series, he walked 5 times (once per 6), but also struck out 5 times. The shift cannot cause strike outs.
Looking at the game situations, there are 3 relevant games: 2, 6, and 7. The Sox won Games 1, 3, and 5, and Game 4 was a 12-3 Cardinals blowout win, but I'll look at that one just the same. And I'll (*) the at bats where the shift may have had an impact (we cannot know for sure, as ground outs happen even without the shift).
Game 2: The Sox managed only 4 hits off of Harry Brecheen in a 3-0 loss. Williams at bats: groundout (*), K, lineup to 2nd (*), foul pop out. The final out was costly, as the other DiMaggio singled to lead off the bottom of the 9th. It was also the only at bat in which a runner was on base.
Game 4: Walk, Single (run scored), groundout (*), groundout/fielder's choice. This last ground out was to 3rd, so the shift had no impact. Sox were down 7-1 by the time of Williams 3rd at bat.
Game 6: Walk to load the bases in the first inning, but was out on a subsequent GIDP, pop fly out, K, single. Sox lost 4-1.
Game 7: Deep fly out (runner on 1st), deep fly out, fly to right (runner on 1st), pop fly out. The last out was costly, as Dom DiMaggio was on 2nd, having just driven in 2 runs to tie the game. The infamous Enos Slaughter play would happen in the bottom half of the inning.
So of those 16 at bats, the shift may have had an impact on 3 of them, and it's unclear that any of those 3 outs had a significant impact on the series outcome. 3 pop fly outs, and Williams did not normally pop out. And it's reasonable to assume that at least one of those deep fly ball outs in Game 7 may have left the park had his wrist been better.
So, it was the wrist, not the shift.