I guess I don't think a day in Rome makes any sense at all. It's a lot of wasted time and rigamarole for the briefest time in Rome. You're spending ≈$300 that could go toward a better flight.
You have lots of options, though. You're gonna have a minimum of two segments each trip; probably three. It sucks but you benefit from those direct flights to Asia so ¯_(ツ)_/¯
Option 1 - link to a direct transatlantic flight to Sicily. That means either EWR-PMO on United or JFK//YUL/EWR(seasonal)-CTA on Delta, Air Canada and I think United for the seasonal EWR flight. That gets you a nice long transatlantic if you can upgrade to business for that segment.
Option 2 (this is almost always what we do) - work backwards from either PMO or CTA. That's what you're doing by routing through FCO in your initial question but I would open your options to other European hubs. Use Flightsfrom.com to search for all of the options. FCO is there with 15+ flights per day but there are a lot others. We often route through MAD or BCN because Iberia has great business class deals and there are multiple daily discount flights to Sicily. I don't worry about flying on a single ticket. I understand the reasoning but that really limits your options and subsequently drives up cost. Discount flights are cheap and ubiquitous so I don't really worry about it.
Option 3 (admittedly this is a bit out there) - Fly into NAP, visit the archeological museum or grab a pizza as you adjust to jet lag, and then take the overnight ferry to PMO. It's the cost of a moderate hotel and it's really enjoyable.