I remain firmly in the “it depends” camp. 
But I thought that the IB coordinator at our school presented some pretty compelling benefits to participating in that program beyond your school (and to that extent, has some similarities to AP). (Note that the IB is quite different from the AP in that there are a number of assessments throughout the year, including papers and oral presentations, that factor into the grade – not just a few hour test at the end of the year.) With that said, he made the point that having top educators from around the world reviewing and assessing what constituted excellence in a curriculum for global students generally resulted in a better curriculum than leaving the decision up to the teachers in a department at a single school. He further commented that the biases of a teacher – be they toward the content or toward the students – tend to be minimized/eliminated and that having outside evaluations brings objectivity and integrity to the process. I DO think that part of it can be scary for some teachers and students. To me, it felt brave to say “We’re going to prepare our students to be evaluated by someone anywhere else in the world with the expectation that they’ll do well in that evaluation”,
And while I appreciate several of the arguments that have been put forward, there need not be an arms race for APs. Our school didn’t weight and didn’t rank – every student was expected to push themselves as much as a they could in whatever way seemed right for them – and that really looked different for different kids. The schools can do a lot to manage whether kids are taking classes to “compete” or to learn.
To say that we don’t want our kids to have AP classes because they could take them at the LPS completely misses the point of why we send our kids to BS … Even if the kids are working toward mastery of the same material (and hopefully, in some of the subjects like calculus and science, they are!), the classroom experience is likely to be very different. As is the residential experience.
So to the OP, whether a school does or does not offer AP classes, I think what you really need to do is listen to each school’s rationale for how they have designed their curriculum and what the intention behind that is. For me, the comments above about the IB felt really authentic to our school’s mission and ethic.