Kyushu self drive December

Hi, we have 4 adults and 2 children age 8, doing a self drive trip in Kyushu this December. We are into nature sights and food but not into temples. Can anyone comment on our trip plan?

Day 1 : Reach Fukuoka drive to Yufuin, overnight in Yufuin onsen
Day 2 : visit Kokonoe bridge and Nabegataki waterfall
Day 3 : hike Mount Aso
(Day 2 & 3 we hope to stay in the same hotel/Ryokan, where do you suggest?)
Day 4 : visit takachiho gorge, then drive to Miyazaki
Day 5 : visit Aoshima island the drive to Kirishima or Kagoshima
Day 6 : take the 4 hour ship (with our cars) to Yakushima
Day 7-8 : explore Yakushima
Day 9 : Kumamoto
Day 10 : Fukuoka
Day 11 : fly home

Please advise does this sound like a rushed trip? Is Miyazaki stop too far out? Any other suggestions for stop over en route from Mt aso to Kagoshima?

I searched in the forum for other similar trips but most people include Nagasaki.. which is more into buildings.. we are into the great sceneries.. so we decided to skip Nagasaki.

Any advice will be greatly appreciated!!

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/33DLp3c
https://bit.ly/2P1k4EZ
A few other things....
For Mt Aso the rim area is currently closed. You can check its status at
http://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/X3LDaE
http://www.thejapanfaq.com/fukpic.html

Kurokawa onsen for your ryokan 2 night stay. Has a big booking website. Lots of info to make your own choice.

Thanks Ken and Hakata14

Hakata14, Will be searching for a nice and reasonably priced Ryokan in Kurosawa. =)

Ken, if we were to include Nagasaki, should we go there first, then Kumamoto, then Kagoshima and Yakushima before heading back to Kumamato-Fukuoka? In this case, should we skip Miyazaki? Miyazaki seems to have some nice natural spots too.. torn between Nagasaki and Miyazaki....

You can order it however you like, though going from Oita to Nagasaki would be very long and tiring - it would be better to go there from Fukuoka or Kumamoto.
As for skipping Miyazaki, its main attraction is its coastline. Which look nice, though Unzen has the mountains, volcanic scenery, Hells, and more hot springs if you like.
https://bit.ly/2OX6sur
So apples and oranges, really.