Yeah, it's certainly a tourist-heavy economy, but they're not letting most Americans fly in at all. I'd think trying to make an exception for just the private jet crowd would be a bad look.
If that were the case it would be hilarious. Nello Musumeci, Sardinia's governor, just last month wanted Sardinia to be COVID-19 free and did not want anyone from any other Italian region without quarantining or constant surveillance.
private jet crowd is likely to spend a lot more money than the commercial jet crowd. after all, they just dropped about $40k or more just to get there.