If he doesn't give the Sox power then compare our number 3 hitter vs the teams that were the absolute best the last two years -- Houston/Altuve, LA/Bellinger/Cub/Rizzo and it comes up absolutely short. The number 3 hitter - which gets the 3rd most at bats-- and the comparison of hitting at this spot is huge.
I feel like this conversation would advance a bit if you acknowledged the arguments from
the Book. But as I'm not going to assign you homework, I'll explain the argument in a bit more depth than I've seen it in the thread.
You are absolutely correct about number of plate appearances. The 3 spot gets ~20 PA more than 4, and so on down the line. But Tango, Lichtman, and Dolphin create a table of the run values of each kind of event (1B, BB, HR, K, triple, etc.) for each lineup position,
adjusted for the historical norms of how often different baserunner/out configurations landed on that lineup spot. Hitting a single with no outs and a man on second is more valuable in terms of run scoring than hitting a single with the bases empty and two outs, right? They're factoring in the historical frequency all of 24 Base/Out configurations for all nine batting order positions. Eight base states — bases empty, man on 1, 2, 3, 12, 23, 13, 123 — multiplied by three out states — 0, 1, or 2 outs.
This analysis suggested that the most valuable spots for the top-three hitters in a lineup were 1, 2, and 4, with 3 and 5 a bit behind. 3 has more PA than 4, but less favorable base/out states. They determined these by running simulations with different batting orders. And it turns out that the difference between their best lineup and the worst (pitcher hits leadoff or whatever) ends up being only 15 runs
These are minor differences. Switching the 2 and 3 hitters so that the better hitter hit second in their optimal lineup earned 2 runs
over a season. But more important than that, they also found that "second leadoff hitter theory" is real: i.e. putting a decent hitter 9, say, JBJ, instead of, say, Sandy Leon, is absolutely worth doing, because it increases the odds that a runner will be on base for your 1 and 2 hitters. This has a season-long effect of about 2 runs. In our situation, I think it's an even better bet for us, because we will likely have decent power in the 1/2 slots. (It's more ambiguous in the NL, where the pitcher's spot is a mix of bad-hitting pitchers and decent-hitting bench players.)
I agree with you that this isn't what all teams do! But some teams do: Kris Bryant is the best hitter on the Cubs — yes, better than Rizzo; he hits 2nd more than any other spot. Mike Trout is the best hitter on the Angels; he hits 2nd more than half of the time. And of course, this isn't an iron law: considerations of handedness or where players are comfortable should absolutely be taken into account, because we're only talking about small differences.