If you are driving all over Kyushu, one of the things you should be looking at is an expressway pass.http://global.w-nexco.co.jp/en/kep/
As for the travel pace, you know your own travel pace best, but don't underestimate the time it takes driving. Speeds are slow compared to many other countries. Going from Aoshima to Kirishima doesn't look very far but will take you 90-120 minutes.
A couple of good places you might like to see after Aso is the Kyusendo Cave - one of the best limestone caves on the island. It's like entering another world and your kids especially would likely love it.https://www.kyusendo.jp/
Also there is Sogi Falls, one of the best on Kyushu.https://bit.ly/33DLp3chttps://bit.ly/2P1k4EZ
A few other things....
For Mt Aso the rim area is currently closed. You can check its status athttp://www.aso.ne.jp/~volcano/info/
It was open for a short time after it was closed close to 4 years. It could open again but honestly don't get your hopes up too high. There are other sights to see in the area and you still can do some hiking there though.https://goo.gl/RwpnU8
You already mentioned Nabegataki Falls which is a great sight. And since you have a car, the Daikanbo Lookout is another excellent place if the weather is good. As for where to say, it depends a lot on your comfort level and budget of course, but if you don't mind another onsen (but quite different from Yufuin), then Kurokawa is one obvious choice. It's outdoor baths can be quite an experience.https://youtu.be/Xs_JXC6bhkM
There are others as well, and not as pricey, like the Aso Uchinomaki Onsen, Tsuetate Onsen, etc. Kyushu has heaps of onsen. For Kumamoto, not sure what you plan to do. The castle is still closed and while the Suizenji Garden is very nice, there is not that much there. I still suggest you look at Nagasaki. The city has buildings (just as any city does) but Unzen is a nice area to see. Lots of nature since that's what you appear to want. The Yutoku Inari Shrine along the way is also one of the best on the island (great ambiance, wonderful mountaintop view).https://youtu.be/2UU_1Aip_l4
For Fukuoka, it has its sights as well - far more than just food as a few erroneously claim. The Nanzoin Temple is wonderful (it's way more than you think), and there are a few very nice traditional gardens.https://goo.gl/X3LDaEhttp://www.thejapanfaq.com/fukpic.html