tank slapper/headshake on scooter

Today I took out the 50cc Honda Metropolitan to charge up the battery
and fill it with gas. I don't usually ride this now days, usually I
let the parents ride it. Throughout the trip I noticed a throbbing
feeling when going 20-35 mph. An up and down motion. The ride also
seemed a bit harsh. On the way home I experimented with taking my
hands off the bars just a little bit, and above 25 mph I noticed the
tendency to enter a headshake/tank slapper.

Remember those potholes you spoke of? Well, they're *hell* on those
teeny tires/wheels....
Check for rim damage.

First, check your tires. Make really, really sure there are no bulges
or other damaged spots in the sidewalls. Damage of this kind can
occur due to road hazards or tire defects and would require immediate
replacement of the tire. Rare but very dangerous due to blowout risks.

Assuming the tires are in good shape, check your wheel balance,
steering head and suspension play and wheel bearings.

Now, when I first bought this bike and rode it I didn't pay much
attention to such things, and it's been ages since I actually rode it
much, so I might have totally missed this. But still, is this behavior
normal, a result of the particular steering geometry of the bike (24
degree rake, 2.2 inch trail, short wheelbase), or maybe a result of a
tire that isn't properly balanced? I also noticed that you can
actually steer the little scooter around just by leaning your body- no
counter steering was needed

FWIW I took my Vino 125 out and did the same experiment. It was rock
steady with hands free riding for brief periods. It does have more
rake and trail than the 50cc Honda, and also hasn't been through 1000
miles of rough roads and potholes.