Sorry it didn't work out, as you probably did find out in the end, buying is just stupidly difficult here in Italy, since a residence permit is required before you can buy and register a scooter under your own name. Plus the prices of classic vespas are incredibly high right now.
On the more positive side, LML scooters, (totally identical to PX series Vespas), are now very common in Europe. Outside Italy, it's perfectly possible to buy, register, (or even temporarily register for export) ,and presumably it's possible to come up with a buy- back agreement from a dealer.
I don't think there are very many rental outfits in Italy who have classic style vespas, and the only ones I've seen (in florence), actually rent LML scooters , and only appear to organize tours, as you said.
If anyone is thinking of doing something like this in future, I'd be glad to give any advice. I'm Italian, living in Slovenia by the way.