I was happy when I started to take the spark plug out and it was loose
I could do it with my fingers, but, after checking the gap, putting it
in tight didn't make it start any better.

Still, the engine on the 50cc motorscooter has started many times now,
but it never runs more than 30 or 45 seconds.

It starts even if no ether is used, often fairly quickly (5-8 seconds
with the electric starter), if it's been 10 minutes or more since the
last time I ran it or tried to run it.

But after 15 seconds it dies.

It runs pretty fast if I give it a lot of throttle, but it still
starts to die within 15 seconds. Sometimes if I had it running fast,
it lopes along very slowly for another 10 seconds, instead of dying
more quickly. If I try to start it immediately after it's died, I
don't think it's ever started then.

Removing the gas cap didn't help (it wasn't on tight in the first
place because I have no key and have to use a screwdriver if I put it
on tight. And I'm sure it's not air-tight even when perfect.)

It couldn't be bad gas or it wouldn't start at all.

I've removed the spark plug and the gap turned oout to be good, 0.7mm,
and it sparks every time, when I electric crank it with the plug out
of the engine.

The owner's manual talks about not starting but not about starting and
dying. I'll go look at the manual

I think it's a fuel problem. It's certainly not compression and I'm
99% sure it's not spark.

I think it's starved for gas.

Parts that come to mind are the electric choke, the idle screw, a bad
fuel-pump-like-device, or some obstruction in the fuel line.

The shop manual says adjusting the idle won't make much difference,
and come to think of it, it dies when the throttle is far above idle.

It starts so there must be enough choke then. How long would a
miniscooter take to fully warm and fully open the choke? A lot more
than 30 seconds, right? but to partially open? 15 seconds? Maybe
the choke never opens at all??? And that's why it dies. Then 10
minutes later the engine is totally cold again?

I tried to take off the fuel line to see how much comes out, but I
didn't have the right tool. And now it's dark out. Tomorrow
should be warm too.

I have to read about the fuel-pump-like-device.