How can I make this effect?

It looks like a sound effector of some kind. Added to a layer, with circular masking of some kind.  There is a trick in after effects where you can add a bass and treble filter. Then you isolate the bass. Then you can convert to keyframes. Then pick-whip to the value of the bass keyframes to the scale of the layer of the video. The movement could just be manually animated as well.

I thought FCP and Pr were good at this sort of thing? Is there not a pre-set that just does it?

Hi, it's very quick and hard to see but I would place a copy of the clip above and cut it down to the section you need ,say half a second. Increase the scale of the copy by 15% (?), then add the Strobe Effect. In its settings Strobe Duration 0.05, Strobe Period 0.10 and set Strobe to Makes Layer Transparent.

Play with timings until you get what you need.

I watched as much of that video as I could stand. 

Do you have a time where your abrupt camera movement happens?