Modern Vespa

Lets start from the top. Moto often starts first go in the morning or whenever
It makes it down the road 5-700 metres and then starts cutting out.
the cure to it dying is throttle which you may not need at the time in fact approaching intersections etc.
when you give it stick it will go but using little throttle is when it is dying.
you may get to where you are going and it will either stop itself with a funny noise or you turn it off at key and may hear same " i am not going to start again for you " noise.
could be 2 minutes later or 1 hr later go to start and it does not start BUT it is a combo of I am not doing any thing and I will turn over and then conk out on you if you attempt to use throttle. If you get it to start not using throttle may work for it to purge its little tanty and will stay started and given 30 secs will not cease upon take off. so then you are on the way. if the beast is starting and then stalling A spray of wd 40 in the bay under seat can get it going.
Todays tanty involved absolutely no response from the start switch for 15 min. just nothing. side stand up. brake engaged. all starts attempted with the lights turned off. switching keys helped one time but not a rule of thumb.Full gas tank seems slightly better than last 1/4. waiting until orange light goes off on speedo is the best time to attempt a start. Have had carby cleaned. Have paid too much for under experienced or under equipped mechanics to attempt to fix. the last one had claimed that he used a laptop to do analysis, but when we brought it back in frustration he said the 3rd test is one that will solve it and that he can not do… only vespa themselves so why did he bother and why not explain to us as we paid him.
when it is running it is a gem and I love the bike. Unreliable restarts due to sdesign is a bore and actually creates more problems than it solves. Heres an extra feature - it starts with the sidestand down sometimes I have noticed.

There are two possible culprits:
1) starved for fuel/air
2) ignition is failing when warm/hot

You will have to do a lot of trouble-shooting to narrow down the cause. If you think it might be fuel/air related, then you will have to change the fuel lines, the vacuum lines, the air filter, the petcock, the vacuum line filter, check the tank breather line, check the intake manifold for leaks/cracks, etc.

For ignition, when the engine is warm, remove the sparkplug and check the spark when you turn over the engine.

Have you checked the valve clearances? Unless you are confident they're OK you could be on a hiding to nothing chasing your tail with everything else.

Thanks MV's,
I got enough from those replys to work with. Cheers for that.