We did this exact same itinerary (except in reverse) on almost the exact same dates this past May. I'm very glad we had the car in western Provence; I don’t think using public transit would have allowed us to see as much as we did. We stayed at a great AirBnb in L'Ilse-sur-la-Sorgue that was very spacious, clean, and affordable. We rented our car at the Nice airport and dropped it off at Avignon TGV to make the getaways easier. I’m sure you could do the reverse, with the caveat that there might be a slightly lower probability of an automatic transmission being available if you need one. We visited Avignon one day and parked on the west bank of the river in an RS-recommended lot with a free shuttle into town. That lot worked particularly well, as our next stop was the Pont du Gard (recommended), which is west of Avignon.
Our rough itinerary for those 3 days was as follows:
Day 1: Drive from Nice to L'IslS (~2.5 hours on toll road) in the morning. Leisurely driving tour of some Luberon villages, including a longer stop in Roussillon (known for red, ochre landscape).
Day 2: Morning drive to Avignon. After lunch, drive to PdG (probably top Roman ruin stop of the region). Since you mentioned Nimes and Uzes, one of those might be better to pair with PdG than Avignon.
Day 3: Driving tour of the Cotes du Rhone villages. Even though I took one for the team as the DD, I had a great time. I thought this was the prettiest part of western Provence. We also did an extended stop in Vaison-la-Romaine (furthest point on the loop from L’IslS). That is another good town for Roman ruins. I recommend the town of Gigondas for a tasting, as well as the English-speaking Domaine de Mourchon in Seguret. At the latter you can actually see the vineyards and do a tasting.
I agree that you might want to figure out what you want to see before picking a base. Our town was great for accessing the Cotes du Rhone villages and the Luberon, whereas the PdG was about an hour I think. The drives are part of the fun though.
For the Riviera, we stayed in Nice and loved it. I don’t think you need a car in that area at all, although Antibes is a little further. My favorite part was walking from Villefranche-sur-Mer to Cap Ferrat. Eze-le-Village probably wasn’t worth it, and definitely don’t believe the estimate of 1 hour to walk Nietzsche’s path down to Eze Bord de Mer.
Hope this helps. Feel free to message me directly with more questions.