Right, to a chacham – not a “tzadik” – who will “ikaper” the anger of Hashem… So, if Hashem is angry at a person (and commentaries follow this line), maybe the first thing Chacham could do is figure out what the anger is about and help defend the sick person, possibly by giving him understanding what H’ is angry about. I don’t see commentaries saying that, but this sounds like peshat.
I am not sure if “sick person in his house” is a euphemism, meaning a person himself, or an adult is responsible for the health of his household?
Shenei Luchot HaBerit mentions this in passing when talking about health, starting with that the best refuah is prevention and taking care of his health even in small matters, and, at the end, it is all b’ yadei shamaim, daven, go to Chacham. So, maybe understanding of the Gemora is that you go to the Chahcam if you can’t figure it out and fix yourself.