Modern Vespa

My 2008 GTS never gave me a problem till I moved to where there are no scooter shops. Funny how that works!
Symptoms:
So, it seems electrical.
Starts, then a few minutes later, the battery seems dead-blank screen, no turnover.

Stops running sometimes while riding. Again battery seems dead.
Push it a few minutes. Starts again.

Had the battery checked, seems ok.
Thought maybe the alternator?
Or perhaps something more sinister.
Any advice would be appreciated

Doesn't sound like a battery problem, it's basically only needed to start the scooter. It sounds more to me like a wiring issue: a loose ground, or a wire with insulation scrapped off that is grounding out on the frame somewhere.

The first place I would check is the connector for the stator under the seat bucket. It's that flat brown three conductor connector at the front of the scooter. While you're in there, check all the connectors to fuses, and relays. It might be you are pinching a wire or connection with your weight in the saddle, but not when it is on the stand.

When it has failed, do you hear the fuel pump when you turn the key back on to re-start?

Nope. Nothing is stirring no sound at all.
However, I just remembered that when I took the battery out, the negative sides screw was a just a little bit loose. Maybe enough to affect it.
I am recharging the battery and will put it back together and see if that was it.
If not, I will start checking what you recommended.

In the process of dying while riding, does it sputter or skip a little bit first?
If it's not the loose terminal, you might want to make sure there is no way the positive terminal is able to touch the side of the body, or get close enough to arc, and check the voltage at the battery when it dies.
An ignition coil will sometimes act out like this before giving up the ghost.
If you have one of the Chinese cheap voltage regulators in there, when they go bad, they sometimes take out the coil on the way out. You can tell if it's a Chinese aftermarket regulator because the area that is full of electronics is about half or two thirds the size of the OEM part.

Just turns off like you turned off the key. No sputtering.
So far it has all OEM parts except where I changed something.
Hopefully just tightening the connection will fix it, but I may not be that lucky!

Well, Topped off the battery and tightened the terminal screws and things seem good again.
Not sure if it makes sense, but I will take it, happily.

Exact same problem with my Kymco, exact same root cause and corrective action.