9 nights in Sicily

Are you sure you are comfortable driving in Sicily? I think your originally proposed itinerary would work fine with a car or even public transportation. You arrive in Palermo and there
is lots to see and do in that area. Spend time exploring the city itself and nearby Monreale. As a daytrip find a local tour or get a private driver to take you out to Segesta and I would combine that with a visit to Erice. An alternative daytrip to Cefalu is easy by either train or bus. Since you really like ancient sites, you could plan to stop at Villa Romana del Casale on your way to Siracusa from Agrigento. While in Siracusa, stay in Ortigia, so very pretty. I know Taormina has become very popular lately and I think you would very much enjoy the Ancient Greek theater there and on your drive there you will have great views of Etna and the coast. Watch for the Cyclops rocks off Aci Trezza.

Maybe cut Palermo to 3 nights and add a night to Siracusa. I would also cut one night from Taormina. Beautiful town, but crowded , expensive and easy to see the sights in one day. I shuttled out to the airport in Palermo to pick up my car. Location is out of the city and easy access to the highways. Driving was fairly easy to Agrigento, Modica, Ragusa, Villa Romana del Casale (don't miss this on your drive east), and parking in Siracusa.
You can get by with one night in Ragusa or Modica (I stayed in Modica, smaller and less crowded). Driving to Catania's airport they morning of the flight could be stressful. You may need to spend the last night in Catania after dropping off the car.

Taormina is beautiful, that said, we saw enough of it for the 2 hours we were there. For your number of days, I'd drop Taormina. There is a lot to do in Palermo and SIracusa, and of course with your interest in ancient sites, Agrigento. Could you stop at Segesta on your to or from Palermo?

We felt driving most everywhere in Sicily was easy. Except, we drove in neither Palermo or Catania. After seeing Palermo, I'd be hesitant to drive there.

What would I miss if I just did Agrigento as a day trip from Palermo and used the train instead of a rental car, just staying in Palermo and Siracusa?

The train from Palermo to Agrigento is 2 hours each direction. Then it becomes personal preference. It's not a day trip for me to sit on a train for 4 hours seeing the same things twice. It doesn't take us long to pack up our belongings and move locations especially with a car. So, for us, we'd stop for a night. Additionally, we spent an entire day at the ruins. I'd worry you wouldn't have enough time.

I think dropping Taormina in favor of two additional nights in Siracusa (Ortigia) would be best. Ortigia seems to be universally adored on this forum, which I wholeheartedly agree with. It’s beyond charming and there is plenty to do.

I think 4 nights in Palermo is good, daytrip to Monreale using local transport, don’t drive. You could conceivably cut it to three nights, but plan this around Monreale’s visiting days/times. Personally, I loved Palermo and could walk it all day.

It’s a drive to Agrigento, so a day trip is kind of counterproductive. Treat it as a leg in the journey across the island. Villa Romana del Casale is amazing and you should consider trying to stop there. This might be a real stretch and a long day unless you stay a night nearby. While the nearest town of Piazza Armerina was a decent place, it’s hard to justify a night off the agenda. But when are going to come this way again?