Driving in Turkey

Many travel insurance policies will have rental car insurance as either a standard or optional benefit. You might also check your credit card to see if it includes any rental car coverage.

In either case, I would confirm that the coverage includes Turkey.

Which company are you renting from? Try to use international brands like Enterprise, Hertz etc, they're available in Antalya. Rentgo is a reliable local brand as an alternative. I'd try to avoid no-name local rental companies.
I've rented car multiple times in Turkey, there never was a minimum 3 days limit and extra insurance was always available. In fact I just checked from Enterprise and one day rental is available. Third party liability insurance is included in the cost but you should be able to buy extra for your car.

As long as you have your driver license you can drive anywhere in the country. It is recommended to resent a car from an agent for the insurance

Just a word of warning. In the US we think of driving down the road as a linear process with set lanes. Driving in Türkiye is much more spatial . If there appears to be a space or the possibility of a space, a car will head for that spot. Having just spent 2.5 weeks in Türkiye, 2 of them driving around the country in a bus tour, I would never attempt to drive there especially in any city.

I wonder if hiring a car and driver for the day might be a better option than renting a car for 3 days?

I rented a car in Antalya last year. I went through rentalcars.com and bought their insurance. Not sure whether that was the right move financially, but it wasn't particularly expensive.

I don't know anything one way or the other about penalties for returning early.

I didn't have any issues driving in Turkey, and I would rent a car again if the need arose. The roads are good and the drivers aren't crazy --- at least compared to Atlanta. There are a lot of good roads leading out of Antalya, and you shouldn't have any issues there. The cities are old and dense, and so pack your patience. Also, parking can challenging. Downtown Burdur was not my favorite place to drive, and I imagine that Konya will be similar. It was okay, just slow. (By the way, just for the record, I would not drive in Istanbul; it's just too dense for me to feel comfortable driving.)

We rented a car from Enterprise this past August in Dalaman. We chose Enterprise because we have rented from them in the States before. We bought the insurance they offered. It was all a pretty easy process. I would suggest using a car rental company you've used in the States, but please make sure you are covered by insurance.

Do we get special coverage from our US insurer?

That is not an option.

We are being told that we cannot rent for one day, we must rent for 3
ALSO! We are being told there is no insurance available

Who is telling you this?

We booked Turkish Airlines. Rented an Avis car through the airline app. It included theft and collision. It was about $200 for four days in Goreme, Cappadocia last week. Driving is a little like Hunger Games, with stop signs and red lights a suggestion and less a rule.
For one day I agree that renting a car and driver makes good sense. Enjoy your trip, not endure it.

I rent a car in İstambul to roam around the city and it was so difficult to park and enjoy it . It is very busy it is better to walk around or use the tram

Just did this route last week. No issues with early return, but take photos of any scratches before leaving. Insurance was basic but our travel insurance covered the rest. Konya is about halfway and makes a perfect lunch stop - took us around 6 hours total with breaks.