It turns out it was a faulty Immobilizer Antenna. Purchased a new one ($165 USD) and replaced it.
I started it up with no issue and let it idle for a while until the cooling fans came on, then test drove it.
I then preceded to turn the key off, turn back on and start, no issues apparent. Took it for a further test drive to keep it hot and still no issue after.
UPDATE:
2022 Vespa GTS 300 Super:
After starting and riding with no issues, then turning engine off….
Immobilizer will activate and will not recognize key when engine is hot (immobilizer light blinks twice then stays on happens with all keys)
This can happen when engine is hot or even if left in the sun all day.
Again, you can start it and ride it with no issues, turn the key off, then immediately turn the key back on, and then the immobilizer will light up and not recognize the key. Even the brown programming key after it is warmed up.
Waiting a few hours or the next day when the engine is cold, it will start up with no issue at all 
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Hey all, first time posting with a heartbreaking issue…
Need help with a peculiar starting issue:
2022 GTS 300 Super, bought used a year ago
At random times after being ridden and the engine is warm, or even if parked in the sun for a while, there is no response at all when trying to start.
Then after even only 15 minutes or longer, it will start right up with no issues. Sometimess just leave it sittinging there, other times after walking it and shamefully pushing it to a quiet corner of a parking lot.
I have changed out to new OEM parts: battery, starter button, kill switch, starter relay. And yes there is power, brake lever held, kill switch off, kickstand up.
Otherwise It starts up and rides fine. The only other odd observation is after riding for a while then shutting it down and parking, there is an interesting sound that sounds like air or fluid going through a struggling pump. Like a whining sound.
So bottom line, after it apparently warms up, it randomly doesn't start at all. Everything seems absolutely normal, but when pushing the starter button there is absolutely no reaction… No clicking, no cranking, NOTHING.
***Again, it can be running for a while, then I can shut off the key, turn it back on, push the starter button and absolutely nothing happens.
If you made it this far thanks for hearing me out. And I welcome any positive thoughts and suggestions from experiences you've had.
⚠️ Last edited by Echo5Victor on UTC; edited 3 times
