Modern Vespa

hi there guys, new guy here. im in Sydney, AUS. I'm hoping to get some advice about a 2013 Vespa i recently bought if possible please!
im not sure what to do as it is an intermittent problem.
after 100 trouble free kms, just a block away from home after a cold start, it stalled when coming to a stand still, but i was able to restart it right away but then it happened again at the next block. then, everything fine for about 400kms, up until today.
it stalled again when coming to a stand still again at an intersection, after a longer ride, but this time i couldnt restart it right away so i turned the key off and on. dash lights were on but it didnt crank over, nothing at all. tried this a few times but nothing..so I pushed it on the footpath and about 30 seconds later i tried again and it started as usual and i rode it home 10kms with no issues.
I had it checked by a vespa dealer after the first incident but they couldnt find anything wrong and they reset the TPS.
FYI, the GTS is 10 years old but only had 800 kms on it when i bought it. it was serviced just before i got it and has a new battery, new oil, coolant, injector cleaned, fuel replaced, tyres, and im not sure why the waterpump was changed. maybe it had some corrosion and was stuck?
it was idling just a little rough so i added a fuel injector cleaner about 50 kms ago, thinking that the stalling and the slightly rough idle may be related.
what puzzles me is that it wouldnt crank over today which it should even if it had no spark for example, or a fuel issue, right?
Ive been reading through several stalling issue GTS posts but havent come across anything similar!
thanks so much for reading. hope you're having a good day or night, where ever you are.

Sounds like the fuel pump to me.

You write that in case of an unsuccessful start attempt "nothing at all" happens, that means the starter doesn't turn? Correct?

I was told a lot of things by a lot of experts, even methods on how to trouble shoot the problem - most of the people had great ideas, but, I agree with the above post - Fuel Pump.
I bit the bullet and had an authorized Vespa diagnose the problem, the said fuel pump and replace the fuel pump - my older 300 has been running great since the pump replacement.

I know your list includes new battery, but I'd start there. Check the voltage and be sure your terminal bolts are tight. Put a 10mm wrench on them.

stall on cold start and able to start again doesn't sound like fuel pimp to me.

dodgy kill switch could cause this. And bad battery could.

Hello Vespa pilots, thank you kindly for your responses. Yes correct, there was no starter turning/engine cranking, just the lights were on. Reminds me of a faulty starter motor solenoid on my old Alfas. Had to give it a knock with a stick and hammer to get it going.
I did check the battery terminal connections after the first incident but I haven't checked the voltage so will do that tomorrow. I will read up to seeif I can troubleshoot the kill switch myself or if it is a matter of just replacing it like the fuel pump, in which case I have to make a decision on which to do first. Thanks for taking the time to respond. I will report back re battery voltage.
Greetings from down under ✌️

Then there is an electrical problem (kill switch, cable break, ground fault, etc.), I do not see a defect in the fuel pump. Measure voltage.
Switch kill switch quickly several times, remove + clean contacts or replace if necessary.

The relevant manuals are here: https://manuals.wotmeworry.org.uk/Vespa/GTS300%20Super/

thank you very much, Jim.

multimeter is busted so havent checked the battery voltage yet. Been reading about kill switches in the meantime.. hmm, the vespa was parked outside for most of its life.. sea side suburb

Vielen Dank, GermanGTSdriver , im on it.

Finally got a new multi meter. the voltage is 12.97 and 14.30ish when idling, thats good, right? the battery cables are thight.
ive been out riding a few times and everything was hunky dory..

say it was a faulty kill switch, would it be a coincidence that it only happens when idling?
if it was a fuel pump issue, the motor would still be cranking but not firing as no fuel from faulty pump. but it wasnt cranking, therefore not a fuel but electrical issue, correct?
need to look up how to get to kill switch contacts- thats next.
thank you, folks.

👍 👍 Yes! Yes! (can double-checked: Do you hear fuel pump for a few seconds when turning key to "on"?)
If you want to access the wiring, you have to remove the handlebar cover.

GT/GTS: Headset Cover Removal

Thank you for the link, touring 300 and the advice, German gts driver. I haven't a chance to remove the cover and inspect the contacts in the kill switch yet, just wanted to say thank you very much. I'll report back with any updates.. hasn't stalled again yet.