your forgetting that he was trying to avoid letting his family know, declaring it in public ruins that, so instead he told her in private
Indeed, this is not a point to be forgotten as it highlights his stupidity even more.
His family needs to know as he's the heir. His family needs to know as the father is the lead pharmacist in the empire.
He's so full of himself that he decided by himself that he was beyond knowledge and saving.
Even if the disease is unknown, this doesn't mean that there aren't ways to research it and find a cure.
In particular, he didn't know about his brother's skills until just now, but he should have known about the empress situation and how she got saved from a supposedly incurable disease. So, how does he not think that
someone had skills beyond his own?
But that couldn't be the case because he, the high and mighty Palle, couldn't think of a solution, so nobody else could.
So, he's going to kill himself (cause of death: arrogance), and keep his family in the dark so they can't prepare for his upcoming demise. And certainly not try to save him.