You make a good point that the defense will likely paint that story later, and it helped remind me that so far, contrary to what the "prosecution" has brought up so far is a bunch of witness testimony that helps the defense, the defense hasn't even started.
On further thought, it actually makes more sense to let that go for now, not push at Gaige's true intentions in cross, because he could very well blurt out, "I was trying to disarm." Rather, it is probably more clever to let that go for now, but wait for closing to point out Gaige's true actions of running after Kyle when he explicitly is heard on video asking what he's doing, Kyle responds, "going to get police," but Gaige chases him instead. Without testimony contrary to that, and without a possible refutation, the defense can now simply point out that Kyle was running toward a wall of police, Gaige chased him instead of rendering aid to anyone, and very likely since Kyle was headed straight toward police, in plain view, Gaige apparently tried to exert vigilante justice; closings are just a summation and some inference can be made from what was presented so far, and neither Gaige, nor anyone else, can refute during closing statements.