Story may not be a prototypical number 3 hitter even with the new philosophy on batting order, but "that's how it's always been done" is almost always a poor counterargument.
The argument for a hitter like him in the spot is he does represent RHH power, and the #3 hitter is the hitter who comes up the most often with no one on base and two outs. The argument against is, that hitter will be third on the team in plate appearances. Story has not done much since coming to Boston, mostly due to being severely limited with playing time, but he was a career .272 hitter with Colorado.
For this Sox team, I'm not sure who is a better option, especially after LHH Devers. Yoshida is one guy who might be able to take that role despite hitting left-handed, but he would have to return to being the hitter he was last year between April 23 and July 25 when his OPS was around .940. That's a good prolonged stretch of offensive success. Of course can he do it again and for longer? TBD. He's not off to a great start. Of course, on last April 18, his OPS cratered to .560. So hopefully he is due for another good stretch. Beyond him, not sure who else...