The fake method you describe is actually pretty close to physically accurate, with a few catches. Specifically, the wavelength of a light effects the IoR of the medium.
IoR is higher when transmitting blue light than red.
One pitfall is that at least for physically accurate renderings, the amount that wavelength impacts dispersion is not constant across mediums. So an accurate dispersion shader would need an IoR and a coefficient.
One example of how different types of glass or transparent mediums are rated for dispersion is with Abbe numbers. An Abbe number, much like the RGB fringe method, looks at the IoR at 3 wavelengths to approximate the dispersion, then just kinda fills in the dots in between. The fringe method doesn’t have a way to fill in the dots without doing loads of extra samples, so the illusion breaks down with high dispersion.
The lower the Abbe number, the greater the chromatic aberration, and generally the lower the optical quality of the medium - a very important factor when constructing things like lenses for cameras and eyeglasses. And on the contrary, when making chandeliers or decorative leaded glass (like for doors and windows) very high dispersion is often desired.