JBJ wasn't even that streaky. He really only had 1 bad month. His 2nd worst month he had an OPS of .780.
But why not Xander? He'd be a bit harder to replace since Ben10 and Betts allows you to sign any OF and stick them in LF. However, he'd also get you a better pitcher than JBJ would and you'd be able to make up some of the production lost by having a better defensive SS. How's Dubon's glove? I wouldn't be eager to trade Xander or JBJ, but if the right pitcher came back... Betts is off the table regardless and is in an entirely different category than JBJ or Xander. I'd guess they'd get 90 cents to the dollar trading JBJ and $1.10 with Xander.
Bogaerts is at worst, an average defender. So a significant improvement with the glove is not going to be terribly easy to find... at least not one that is coupled with a bat that doesn't make the drop of a significant negative overall. He had an .802 OPS with at 113 wRC+ this year. That's the 6th best shortstop bat in the majors and 3rd best in the AL. And he still has plenty of room to grow.
Now let's look at their options for replacing him:
Internally:
MLB Experience: Brock Holt, Deven Marrero, maybe Michael Martinez or Marco Hernandez?
None of those are serious options. Let's move on.
On the farm: Mauricio Dubon, CJ Chatham?
Dubon may have just had a breakout season with half of it spent in AA hitting to the tune of a 151 wRC+. Even still, soxprospects has him at 10 on their top prospects list, behind Marco Hernandez. That's probably because his defense doesn't profile as any better than Xander's while having a bat with a much lower ceiling, even if he has a chance to be a solid hitter. He's not an awful option, but he wouldn't be ready until late 2017 or early 2018 so they'd still need to pick up a replacement externally.
Chatham could develop into a long term replacement, but having just been drafted, he's at least a full season away from any kind of serious consideration and could easily flame out. He had a nice debut, though.
Externally:
Free Agent Market: Erick Aybar, Alcides Escobar, Alexei Ramirez, Eric Sogard.
That's the entire free agent class. Again, let's move on.
Trade market: Eduardo Nunez, Zack Cosart, maybe Marcus Semien or Danny Espinoza? Elvis Andrus or Jose Igelsias as a long shot?
Nunez wouldn't be an awful option, but would probably be a much better investment as a utility bench player. Cosart plays excellent defense but that bat is probably never going to come around. The A's have little reason to deal Semien. Same with the Nats and Espinoza, but Turner's emergence might make him attainable. And maybe you get lucky and the Rangers want to go all in on Profar and would dump some salary in Andrus, who is solidly average still. Or the Tigers are willing to punt and will package Iglesias back to the Sox? None of those names are star players, nor do they have the ceiling to hope for that, though. And any of these guys will require moving more prospect capital to bring back.
So even if we concede that Bogaerts is going to be worth more to Chicago (or whomever) in a trade, replacing Bogaerts right now would not be easy and would likely be costly, minimizing the impact of whatever trade he's involved in in the first place. Trading Bradley doesn't require a lot of movement to fill the gap he leaves behind. With Betts, Benintendi, Young, Swihart and Holt in house, and Desmond, Bautista, Reddick and maybe Cespedes on the free agent market, they are almost assuredly capable of finding a replacement without further thinning the farm.
TLDR: JBJ is FAR easier to replace at this moment and thus the far more attractive piece to include in a deal for a front line starter.
That said, I'd rather they didn't trade for another front of the rotation type guy and kept the current outfield in tact. Of course, I don't buy into the idea that Price won't ever be able to pitch in the post season, which helps me to be comfortable with that.