Jason I am sure that the caps can and do work in light situations and used a science test ok....BUT...
That is really not the best solution, and for proven reliability we use higher capacity CCA, AGM or lead acid batteries for our cars. I would not want to chance problems by even trying the connections with caps. I cant afford computer related problems/nightmares.
In the video above with the car running, with the battery terminals disconnected, I am not so sure that is a good idea either..for the alternator alone.

I'm lost, what is the supposed benefit to capacitors in this application?

Only advantage to capacitors here would be that they are far lighter than our lead batteries.

But, if these capacitors truly did work, why didn't GM or Ford or Toyota use them? I still think the video is a fake.