Recommendation on PUglia Base?

With a car I'd be inclined to split my time between two bases to reduce driving time.

I liked Lecce a lot. It has a lovely Baroque historic center and feels lile a real city, not just a tourist destination. But it's not a small place, which might present parking and navigation issues. Bari's even larger.

Some of the coastal towns have white-washed cubic architecture somewhat like Greece, so their atmosphere is quite different.

I don't think too many visitors opt to stay in Alberobello because it's very touristy--though my guess is that the crowds are mostly daytrippers.

I'd weight geography and driving time heavily in this decision. Use ViaMichelin.com for estimates of the latter.

we stayed in Trani and thought it was delightful. While surrounded by a bit of a suburb, it has a great Old Town area that is very walkable. Polignano seemed nice, too, we visited as a day trip. We also visited Bari and thought it was pretty busy; nice for a visit but wouldn't be my first choice of a place to stay since it is a ferry port so a somewhat busy town compared to other choices. Given the distances up and down the coast, it might be better to spend 2 nights in one place and 3 in another.

We spent a week in Polignano a Mar and loved it there with it’s waterfront location and old town. We toured all over and even did a day trip to Matera. Lecce is beautiful but not directly on the sea. Monopoli is very near Polignano, also on the water and another good choice. We liked Trani a lot too. Make sure you walk around Bari’s old town and visit the cathedral.
Being on the waterfront is important to us.

We stayed in a few towns in Puglia. I agree Lecce would be a good base. From there we visited Gallipoli, Galactica, and Otranto, and drove south along the coast. We could have driven to other towns but we changed locations instead. We also stayed in Ostuni, a 'white town' and visited other towns from there. Bari is also a possibility; on the sea with a beautiful old town. We dropped off our car when we arrived and stayed near the historic center. Everything was within walking distance.

I agree that you should list the places you want to visit and check the driving times to help decide where to stay.

We've stayed in many of the above mentioned towns, except Gallipoli and Polignare al Mare, and I don't think you can go wrong with any of them. I also vote for Lecce, with it's beautiful Baroque buildings, great churches, and a sweet little museum at the Roman Theatre that marks the end of the Appian Way. We were there mid-week, and had the entire place to ourselves for 2 hours. There's also a neat little museum, Faggiano, if I recall correctly that was once a family tavern that turned out to be a great archeological find when the owners started digging around to do some renovations. The food and gelato were wonderful. Lecce old town is lovely, but parking can be a pain. The owners of the airbnb where we stayed included parking in a nearby lot; we were inside the gate (Arco di Prato), but the car was outside the gate. We stayed in Alberobello, mostly because we wanted the experience of staying in a trullo. I don't know if it was our particular property or not, but we found it cold and damp. However, the owner's gourmet meals and willingness to do our laundry (for a fee, of course) while we toured made up for that. We were there in mid-October, so the weather was cooler. In the heat of the summer, I'd bet they're like living in air-conditioning. (We got a parking ticket there, which was the beginning on an 18 month journey through the Italy's police departments and Europcar's admin department. The Italians were easier to deal with, but I digress.) From there, as I recall, we did a day trip to Polignare al Mare and that was enough. I didn't really like Bari. To me, it was overly congested, dirty and I never really felt safe wandering around the old town. It's also a port where some smaller cruise ships call, so check the schedules if you're planning to stay or visit. Adding 2,000 - 3,000 people to the mix would be really unpleasant. Ostuni is strikingly beautiful, but a little dull after dark. Couple of good restaurants, but not much else. What it has going for it is plenty of free parking, albeit at the bottom of a staircase of about 100 steps. (Maybe there's a shuttle now; there wasn't when we were there.) There was also a few pay parking spots at the top of the hill if I recall correctly. I think choosing one town over another depends on your itinerary (in order to pick something centrally located), parking (always a consideration for us), and atmosphere.

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Map the drives and determine how much you like to drive in one day.
Lecce is the most convenient for visiting both Otranto and Gallipoli, so you might pair that with a central location for the other few places.

From all the research and the RS forum contributors, it seems Polignano a mare 2 nights and Lecce 3 nights seem most convenient.
Any hidden gem spots I missed?

While exploring the Itria Valley around Alberobello, be sure to check out all of the little towns--Locorotondo, etc.

I spent one night in Alberobello. It was a five minute walk to the central part of the Trulli's. Crowded during the day but peaceful at night

I also liked Lecce and based there a few days. I made day trips to Otranto and Gallipoli. I only spent a few hours in Polignano but enjoyed it. This was not a day trip but on my way out of Puglia.

My trip was in May of last year. Outside of Matera and Alberobello, Puglia was not that crowded. And few Americans.

After Puglia I went to Sorrento and Amalfi where it was wall to wall tourists. (This was a return trip 28 years later.)

I aree with those who suggest 2 bases. The driving can get really tiresome. In Lecce, we stayed at a b&b just outside the wall of the historic area. 5 minute walk to get in, and everything inside can be reached on foot. Bonuses were free covered parking, wonderful breakfast, and delightful owners.
Any of the towns of the Val d’Itria makes a good second base. You do not need to stay in, or even visit Alberobello, to see trulli. They are all over the Val d’Itria.