Best of Turkey Tour, what to plan myself?

I highly recommend, if it is offered, to take the pre-tour foodie tour with SRM (if available it would be noted in your tour documentation). It's the day before your tour departs and is a wonderful way to sample food and get a good orientation to Turkish culture and cuisine. I also recommend arriving two days before your tour starts so you have one day to acclimate and see things not covered there, and one day to do the food tour. Simply not enough time in Istanbul to see everything.

As you noted, free time could definitely vary depending on your tour guide and the day your tour starts. On my BOT tour in late September, the first day we met around 1 p.m. and did not stop moving until nearly 10 p.m. There is a lot of bus time, with frequent rest stops but not really long enough to do extra things. Of all 5 RS tours I've taken, this one had the least amount of solid free time. Anyway, here are a few thoughts:

Istanbul: At Topkapı Palace we got about an hour to wander around on our own. The guide will point out options of things to do, but I went through part of the treasury and then sat down for a rest (lots of walking). We were given a short lunch break on the second day near the main mosques - our guide pointed out options, but keep the RS guide handy for reference. I think we were free for dinner the last night in Istanbul, but I was tired so got a take away dinner and got my bag arranged for departure.

Ankara: you get there at dinner time, sleep, and leave the next morning. No meaningful free time.

Cappadocia: there isn't a ton of free time here, especially if you do the optional balloon ride (DEFINITELY DO THIS). They try to have you do the balloon on the first morning you are there, in case the balloons cannot fly then you can try on one of the next two days. The only thing I didn't enjoy on one of our afternoons was we had a choice of going on a walk or visiting a traditional pottery place - I picked the latter that ended up being a sales pitch (after the carpet sales pitch I was pitched-out). Depending on where you stay, there may not be a lot to entertain you other than getting dinner (the guidebook will have options).

Konya: we had some free time for dinner. I ate with several of our tourmates and our bus driver.

Antalya: this is the stated "vacation from vacation" spot and where you'd want to invest in some brainstorming what to do. I chose to rent a boat and go swimming with some of our group (wonderful) and also go to the Turkish bath (wonderful). Both easily accomplished on the same day. Our tour was there on a Sunday and there were fewer restaurant options it seemed.

Pamukkale: again, not much free time you arrive pretty late. Some of the group swam in the hotel spa pools after dinner. At Hierapolis you are set free for several hours - I chose to explore the site on foot, others chose to go swimming in roman ruins.

Kuşadası: you have part of an afternoon to explore the town on the first night and after you get back from Ephesus on the second day. I had a later return flight than others, so I took a nice long walk before catching a shuttle to the Izmir airport.

I haven't looked at the RS itinerary so I don't know if the Mehter Band concert is included in Istanbul. If it isn't, it is a very interesting cultural experience. Yummy Istanbul's, "Taste of Two Continents" food tour is excellent if you have a full day prior to your tour.

Pre tour add at least two days, more would be better. In Istanbul we went into the Bassilica cistern, which is beside the Hagia Sophia, as well as the archaeology museum. We also walked up to the mosque of Süleymaniye the Magnificent, passing through the spice market and the back alleys on the way. We walked across the Galata bridge, up to the Galata Tower, and explored the streets there, and then walked down the to the cruise port and wandered along the water. In general we just did a lot of walking and exploring, but it was a lot of fun.
As far as the tour goes, there is not a lot of free time as CL mentions, but in Cappadocia definitely do the balloon ride. Some of us did take up the offer to go to the pottery place, also. I really enjoyed this, and yes, they do try to sell you something, but I didn’t feel overly pressured to buy. The pottery is stunning, and the demonstrations were quite interesting. I did buy a couple of small items as I loved the designs.
In Antalya our guide organized a boat tour for a group of us that were interested. This was nice, even if you don’t want to go swimming. Afterwards, some of the group took taxis to a museum, which was quite good by all accounts, and the rest of us just explored the town.
I didn’t take advantage of the opportunities to swim as some did. Instead I took the opportunity to wander and take photos.