Don't overlook the adjustable elbows. Whatever method you chose, (tape, mastic), elbows can leak, especially when stressed, so look at those areas also. And get the dust/dirt off of the area first. Personally, I think both mastic and foil tape look like crap. Aluminum shade silicone caulk can work in some areas, and kinda blends in if you're concerned about those things.
When using mastic, you can apply duct tape to give a mask edge, then remove the duct tape. Leaves a clean edge.