Yes, it's not as if they don't already have their own 1st round pick + the comp. pick for EE. Giving up the 2nd comp. pick by signing JB, who is in real danger of being frozen out of the market altogether anyway, is a reasonable trade-off for adding a 3 WAR player to a team that is still in a window to compete now. Also, if it is only a short term deal, there is an advantage in not having much money committed, in the event he does continue to decline.
I think you are right that they are more concerned about his physical state OR mental state / attitude. Remember the comments by Gibbons in an interview where he said the big turning point for the team was bringing in Martin and Donaldson in 2015 - tough guys with a real will to compete and win. That seemed like an indirect shot at JB, EE, etc. So it may be not only the FO, but also Gibbons who are not too thrilled with JB and may see him as a clubhouse/dugout problem that subtracts from the overall effort, despite his ability to produce at the plate. Also there is his bad relations with the umpires and opponents that greatly affects the team's reputation as a whole. Bottom line: the intangibles that the stats guys can not quantify are likely playing a big role here.