OK, so 12 days total, "door-to-door", including flights. So 10 days on the ground, being there. You need to be in (or at least near) your departure airport on your last day (evening). If you throw in a flight between any two places (eg Rome and Olbia), that's at best 9 full days to spend (setting aside concerns about the day following arrival, which for many of us, is not a day we are at 100%, due to jetlag and travel displacement trauma). Finally, remember that any group only travels as fast (or as happy) as the least tired, least grouchy, least enthusiastic member.
Rome's alleged shortcomings:
- Crowds. Yep. Unless you have a time machine, all one can do is try to mitigate them, to some degree.
- Grungy city. Around the edges, sure. It's a big, old city. There will be garbage, dog poop, people, more. Comes with the territory. Yet people keep coming, for reasons which are probably obvious.
- Not always friendly. Maybe. But like most other cities.
These shortcomings are also true of most big cities in the world, and plenty of smaller places, too, certainly places that are popular with tourists. They're (usually) popular for good reasons though, so you either go and live with the crowds, or you skip these cities, your choice.
Rome versus Sardinia
In Rome, your biggest challenge will be prioritizing and deciding on how to spend your limited time. Many people would go and spend 10 days in Rome happily, not going elsewhere, it's overflowing with 5-star attractions, like few other places. And lots of other wonderful places are just a short train ride away, so plenty of us will combine Rome with other beautiful Italian places. You just have to choose. In Rome, you do NOT want a car, period.
In Sardinia, there's little public transportation that's useful for casual (short-trip) tourists. You will either need to rent a car (easy) or just stay at a resort (a waste of a trip to Sardinia IMHO, but I know, folks want their beach time).
In Sardinia, things are spread out. But you only have time for one base (in/near Olbia). With that base in the northeast, and a car, you could get around to many interesting places on the north coast: the La Maddalena archipelago (involves ferry travel, so costs time), Santa Teresa Gallura, Sassari or Alghero, the peninsula around Stintino. South of Olbia, the landscapes are spectacular, along the Gulf of Orosei and beyond. There are cool little towns and great cities further south. But all of these places require some travel time, and you only have a tiny bit of that to spend.
Splitting one's time comes with compromises. I have learned the lesson (and keep re-learning it) to focus more time on a smaller target. I'd rather spend more time exploring more of one region in greater depth, and defer a trip to another region for a later time, rather than trying to spread myself across many regions and ending up with just a few small tastes of different places. I still struggle with it, as we all do. It always comes down to trade-offs.
Both Rome and Sardinia are wonderful destinations. I'm sure you will enjoy your time in either/both. Good luck with your prioritization..