Question about ideal filters
In other words, $h[n]$ is ideal but its frequency response $H(e^{jω})$ is practical and realistic.

This is not true. To compute $H(e^{j\omega})$, you must evaluate the summation $\sum_{n=-\infty}^{\infty}h[n]e^{-j\omega n}$. A real system can't do this because $h[n]$ is not finite length.
Say you choose to use a rectangular window $w[n]$ (with length $2M+1$) to force $h[n]$ to a finite length and the summation becomes $\sum_{n=-\infty}^{\infty} h[n]w[n]e^{-j\omega n} =\sum_{n=-M}^M h[n] e^{-j\omega n}$. There is still the problem of $\omega$ being a continuous variable. For example, a computer can represent $h[n]w[n]$ in an array, but how is a computer supposed to store $H(e^{j\omega})$ in memory? It should have to store all function values between $-\pi$ and $\pi$ of which there are infinite.