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Let me check this for real matrices and $f(A)={\rm tr}\,A^2$. The $N$ eigenvalues $\mu_n$ of $A$ have in the limit $N\rightarrow\infty$ at fixed $N/K=\lambda\leq 1$ the Marchenko-Pastur distribution $$\rho(\mu)=\mathbb{E}\left[\sum_{n=1}^N\delta(\mu-\mu_n)\right]=N\frac{\sqrt{\lambda_+-\mu/N}\sqrt{\mu/N-\lambda_-}}{2\pi\lambda\mu},\;\;\lambda_-<\mu/N<\lambda_+,$$ with $\lambda_\pm=(1\pm\sqrt\lambda)^2$. The function $f(A)$ tends in this limit to $$f(A)\rightarrow\int_{N\lambda_-}^{N\lambda_+}\mu^2\rho(\mu)\,d\mu=N^3(1+\lambda),$$ which differs from the answer $f(N I)=N^3$ conjectured in the OP.