Recommended onsen near Kyoto or Lake Biwa

Hi everyone
Me and my partner are heading to osaka/kyoto in May and are looking for a ryokan for our kyoto leg. we would prefer one with a private bath, but I am having a hard time searching for one. we are open to exploring other areas near kyoto (e.g. lake biwa). Does anyone have any recommendations for ryokans in these areas? we are flexible with the budget. thank you.

This is a map for onsens providing rooms with open-air baths.
https://www.jalan.net/onsen/heya_roten/WID_35.html

And this is for onsens with private baths.
https://www.jalan.net/onsen/kashikiri/WID_35.html

In the Kansai area, the top two onsens are Kinosaki and Arima. Both are shown in the maps above. In Kyoto city, there is Arashiyama Onsen. Ogoto onsen is on Lake Biwa. This site lists up rooms with open-air baths in Kansai. There are many.
https://www.ikyu.com/dg/special/fullyear/outdoorbath/start5.aspx

From the above, I just pick up the ones with the word onsen in the below.

Kyoto Arashiyama Onsen Kadensho
https://www.ikyu.com/00001842/plan/10010e/?acr=262144&ipb=1
https://www.hotespa.net/hotels/kadensho/

Ogoto Onsen Yumotokan
https://www.ikyu.com/00001839/plan/10010e/?acr=262144&ipb=1
http://www.yumotokan.co.jp

Ogoto Onsen Biwako Ryokusuitei
https://www.ikyu.com/00002098/plan/10010e/?acr=262144&ipb=1
http://ryokusuitei.com

Arima Onsen Gekkoen Korokan
https://www.ikyu.com/00001033/plan/10010e/?acr=262144&ipb=1
https://ssl.gekkoen.co.jp

The private bath at Yomotokan appears a one for one person (see the second photo). Private baths are rather rare at any onsens. You should need to ask any ryokan for the details.
http://www.yumotokan.co.jp/world/english/room/roten.htm#r_nav