See in the video where your friend rides it. As she pulls away, or goes from idle to putting a load on the engine, it sounds like it's not running on all four cylinders. Then once the revs come up a bit it stops hesitating and runs on all four. I have had that problem in the past and what sorted it for me was replacing the spark plug caps. If yours are the original Honda ones from 1989 I would suggest replacing them with new NGK caps. What happens is over time the insulation on the cap breaks down so the spark finds an easier path to ground by jumping from the cap to the head, instead of jumping the spark plug gap. The result is the engine misfires/hesitates/drops a cylinder.
If the HT leads unscrew from the ignition coils you could also buy a length of HT lead cut it to length and replace them too. You could probably get four NGK caps and a length of HT lead for under £20. It might solve your running problem. Even if it doesn't replacing the 30 year old stuff should give you a stronger spark.