score:6
Your reasoning is correct.
Kayseri + rental car is the most convenient option.
You have much more choice to and from Kayseri: better schedules and better prices, through competition. Moreover, the car rental facilities tend to be better at Kayseri. And it is not that further. Anyway, the drive is a quite relaxed one.
Renting a car is definitely a good idea to explore that region. You will be more flexible. Getting from Göreme to more remote places like Mustafapaşa, Ihlara Valley or Kaymaklı is much easier and quicker with a car than with public transport. And you will be at ease to visit Kayseri. Indeed, if you have the time, stop by for a couple of hours. It's worth it.
An alternative would be to use an airport transfer, pulbic transport when possible and eventually take part in organised trips to the more remote areas or rent a car for that purpose on a daily basis. The organisational effort is however much bigger. You also lose some flexibility and spontaneity.
A last word on the airports. Check carefully if your flight to Kayseri is operated from Istanbul Atatürk Airport (code IST). There is another airport called Sabiha Gökçen (code SAW). The latter is 50 kilometers from what one would call the city center. And transportation to and from this airport is by far not as straightforward as from Atatürk airport.