6 days in Sicily should we skip Catania?

As so many people have said on this forum, Sicily is BIG and you are doing way too much!
First your question: I'd skip Catania.
2 days in Syracuse is fine, but with only 6 days I'd stay in that part of Sicily.
We just did:
2 nights in Syracuse (stay in Ortigia).
2 nights in Ragussa with a stop in Noto and Modica on the way there.
1 night in Agrigento to see Valley of the Temples

We were in another part of Sicily for 2 weeks before that and felt we could have spent way more than the 3 weeks we had. So much to see, it's wonderful!

A couple of things come to mind as I read this. I assume you are arriving in Catania from another European destination and not an overnight international flight from America. You say you have six days, but think in terms of nights. Do you have five or six nights? Do you actually have a full six days for siteseeing? Are you leaving Sicily from Palermo or do you have to double back to Catania? Do you feel comfortable renting and driving a car in Sicily? It gives you greater freedom of mobility but do you need to return the car to the place you rented it from? You do not have many days so you are going to have to prioritize. It seems to me that the two days you have allocated for Siracusa/Ortygia and the south region are barely enough to see just Siracusa/Ortygia. There is a lot to see and do there and if one of these days is your arrival travel day you are already down to a day and a half. As much as I personally liked Catania, you will not have time for it or for any other daytrips to the south. You can catch a bus at the airport and go directly to Siracusa and wouldn't need to rent a car in Catania. You could also go by bus or train from Siracusa to Agrigento. Agrigento needs several hours to see the temples, so the travel from Siracusa to Agrigento, temple visit and then the continued trip to Palermo takes up one of your allocated 4 days for Palermo. Could you possibly just rent a car in the Palermo area, (if you do not need to drive back to Catania to exit from there)? So then you have only three days for Palermo. Again lots to see and do in Palermo itself and its immediate surroundings. If you want to do a daytrip you could go by bus or train to Cefalu or possibly join a daytrip tour to Segesta and/or Erice. You probably won't want to drive in Palermo itself.

Another idea that occurs to me if you enter and exit Sicily through Catania, is just to stay on the eastern or southeastern side of the island. Save Palermo and the west for another visit. Maybe combine Taormina, Catania or Etna and Siracusa. Or if the South east attracts you more, then look at Siracusa, Noto, Ragusa. Lots of possibilities.

i think you should decide what interests you most. for me, last year's trip to sicily was about norman & baroque architecture, caravaggio paintings, and few specific sites. i usually avoid beaches, gastronomy/wine, and ancient artifacts. regarding stop-off points, i would suggest piazza armarina and the nearby villa romana del casale. the roman mosaic there is incredible. segesta (imo more picturesque than agrigento), on the way to trapani. erice just above trapani is an amazing medieval hill town with hardly any modern additions. as far as catania, it's a great base if want to go to the small towns surrounding etna via the circumetnea rail line (itself a fun experience). randazzo was my target. i also went to acireale and caltagirone from catania. catania by itself was not as interesting. of course, palermo is a place where you could spend your entire 6 days! there is so much to see in the city itself but also great for day trips to monreale and cefalu. lastly, an added bonus are the genuinely friendly and helpful sicilians themselves! no matter where you end up going in sicily i am sure you'll love it.