Does adding OWF of the private key to encryption scheme hurts security? Suppose I have a symmetric semantic-secure encryption system $\Pi = (Enc, Dec)$ and an OWF $f$.

Now, define the following encryption scheme $\Pi^{'} = (Enc^{'}, Dec^{'})$ where , $Dec^{'} = Dec$ and $Enc^{'}(k, m) = Enc(k,m)\mathbin\|f(k)$

Is that encryption still secure? I already tried to formally prove this claim with a hybrid argument, i.e. claim that

$Enc(k, m0)\mathbin\| f(k) \approx Enc(k, m0)\mathbin\| f(r) \approx 
Enc(k, m1)\mathbin\| f(r)  \approx Enc(k, m1)\mathbin\| f(k) $

However, this proof turned out to be wrong (for example, maybe $Enc$ and $f$ both reveal the first bit of $k$ in the clear).
It's possible to have a secure encryption scheme that ignores the first half of its key, and a secure OWF that leaks the entire second half of its input. Composing them as in your question results in something profoundly insecure.