SICILY: need opinions re part-driving/part public

I would consider to train from Palermo to Agrigento and pick up rental car there and then return it at Ortygia. We did the reverse. We used Avis. We took an Uber from our AirBnB in Ortygia to rental car office just outside of town. Had it for 3 days for the hill towns and Agrigento and returned it near the train station in Agrigento and took the train into Palermo. Our AirBnB in Palermo was walking distance from the train station but Uber worked well there too. We watched some youtube videos about driving in Sicily but didn't find it too bad for what we did.

5 nights in Palermo is probably quite a bit. I presume you intend to take a day trip to Cefalù by train while there.
Personally I would not stay that long in Catania at the tail end. I would stay only the night before your flight out.
I would use those nights to stay, with your car, in Western Sicily after Palermo. In that area you can visit Segesta, Trapani, Erice, Scopello, Zingaro Natural Reserve, Selinunte, San Vito lo Capo, etc. Just choose a base anywhere (I liked Scopello) and drive to the others.
Another option could be a couple of nights in Taormina or nearby Giardini Naxos.
Ortigia, where you are planning 3 nights (=2 full days) could be a base for a day trip to Noto. One day you will need to see Syracuse Archaeological park. Ortigia is small and can be visited in the evenings while you stay there. You can park your car at the Talete covered parking lot, in Ortigia.
I would rent the car on your way out of Palermo and return it in Catania the day you arrive to spend the night before your flight.
Concerns with gasoline situation are exaggerated. Gasoline is not going to run out in the next month. Italy has a huge refining capacity and actually exports refined oil products. Less than 10% of crude oil imported by Italy comes from the Persian Gulf. Half of it comes from Africa (mostly Libya, Algeria, Nigeria), most of the remaining 40% comes from Asia (Azerbaijan, Kazakhstan, principally), Europe (Norway, UK) and the Americas (USA and Brazil). Actually Italy imports more oil from the U.S. and Brazil than from the Persian Gulf. The risk so far is for jet fuel, since Europe refines only 60% of its needs for that product, not cars. In terms of gasoline price, the cost is currently €1.80 per liter (about $8 per gallon), actually less than a couple of years ago, since the Italian government has eliminated several gasoline taxes recently during this crisis. The cost of gasoline, although higher than in the US, will not impact your budget much at all. The distances you will be traveling are not that long. I can guarantee you that your entire gasoline expense will not be more than $150 for your entire trip. That is not even a dinner for two at an inexpensive restaurant in your NYC.

Ortigia, where you are planning 3 nights (=2 full days) could be a base for a day trip to Noto

And/or a trip to Modica. We visited both Noto and Modica on our first Sicily trip, and far preferred Modica, for its setting and architecture, plus the chocolate! Parking in Noto was a pain, too, but easy in the lower town of Modica. Do see both the upper and lower towns.

The Villa Romana at Piazza Armarena was easy by car, so I’m glad you’re not trying to do it by bus. But a private driver vs. driving yourself? As Roberto suggests, keep your car for longer. If gasoline prices and availability were truly life-changing, vacation-altering next month, private drivers would have to jack their prices way up to compensate, so you’d be paying for that either way.