No cus any sane person would've ask for the doctor's help so it's not that important to highlight. Especially if the sick person has their own personal doctor.
And just because Rishar thought the hidden identity is bad for MC's health, doesn't mean it's a clear indicator that he is in love with him? It's sure is a clear indicator that he's a good person tho.
From the butler's pov, they're both good friends but due to his duties, he chose loyalty. Cus again, he lacks context about their relationship so thinking of the worst (rishar using the secret against mc) is the best rationale. Do remember that the mc himself doesn't want the public to know about his real identity + his abusive mom also think the same so it has real consequence to their politics. The butler isn't really one to blame here (not like he can do much other than being loyal/die keeping a secret)
Not gonna lie I get a lil annoyed to butler but his reason were understandable. but bruh take a hint the duke were literally in love to your highness you can clearly look to him his only care in the world were the prince