My son and I are going to Italy to buy

Yes, I'd heard that, but it's obviously not as important as it seems, or some would like it to seem, because non-residents from all over the EU buy cars in Italy and take them home all the time. Re-registering it is no problem. I suspect you might be close to right about the insurance: one probably needs to get a dealer, or the previous owner, to insure the car on a short-term basis until it can be re-registered abroad, which takes about a week here. My insurance company is happy for me to buy insurance here, at this end, so that's probably the way to go. (It's only a couple of euro if I decide not to buy that particular car and cancel the policy a couple of days after buying it.) Does anyone know a bit more about this, or have any other (preferably more constructive than poisonous) comment? I'm really looking for someone who knows what they're talking about here. You've all got cars; someone must know someone who knows something about (buying and selling for export) them. 

Well, perhaps 'poisonous' might not be the best description, but I couldn't help being struck by the feeling that the purpose you intended them to serve when you wrote them was about making yourself feel exclusive as one privileged to have a car in Italy, more than they were about trying to help me get a car there. The links are interesting, thanks, but I've been there before. They're not relevant to buying a car for export, which doesn't (obviously) require residency. The most interesting point was made in that last one, '...buying-a-car-in-piedmont...': "Go with an Italian and don't speak." That's what I'm looking for – an Italian to go with. I thought I'd made it clear from the start. I guess I must just write badly. 

Only if you live there. Neither bicycles nor 'monopattinos' (whatever they are) are as good for your health as they are dangerous when there's ice on the road, as there is for several months of the year here. 

Depends on circumstance, I should think. If you've got four kids in school, all of whom need to be taken to various sports and interest groups, reading and knitting, with or without a fire, may not be an option. I used to take my son to kindergarten by bike a couple of years ago, before he got too big and heavy. (We use a forward-mounted kids' bike seat.) I did over a thousand miles that autumn: 20km there together and 20km home for me alone in the morning followed by the same again to get him back in the afternoon. The weather was good enough (no ice on the roads) to keep that up right through until October that year, although his mother used to make an awful fuss. So, I'm not averse to cycling – nor reading, nor knitting – but we really do have many good uses for a car, particularly during the winter. 

Just couldn't find exactly what we wanted there.

No, EU citizens pay VAT on cars only once – to the seller where it was bought, not in the country into which it is imported. I've already imported a car (from the UK) to Estonia, so that part of the process is quite clear. It cost about €200, but that's only because UK registered cars are awkward: they need to have new headlights (right-dipping) and a rear fog light fitted. State (re-)registration fees c/w plates, which includes informing the other EU state to remove the car from its register, were about €50 (might have been €80 – can't remember) – not a daunting obstacle.