It's the "Are you willing that ..." that is the problem for me. It has been used but is completely eclipsed by "Are you willing to ...". The examples for the "that" version do sound very formal or archaic and it had dropped in frequency from earlier times. (I have a couple of old Japanese dictionaries and they have some "interesting" entries too )
The relevant Ngram speaks volumes, with "are you willing that he ..." scraping its undercarriage along the tarmac. It doesn't fly. The fact that a construction (here, "are you willing that ...") occurs out of context in a dictionary doesn't validate the construction for today's usage.
It apparently went viral in 1651 though!