Yes — this idea is known as pseudo-gray. It uses slightly non-neutral RGB values to create extra apparent luminance steps between adjacent neutral grays on an 8-bit RGB display.

In practice, though, the benefit is limited. The same reason the color tint may be hard to notice is also why the tiny brightness differences are often hard to notice. Human vision is not very reliable at judging such fine luminance steps, and display behavior is not perfectly precise either. So while pseudo-gray exists as a concept, it usually is not worth much effort for normal photo display or editing.

In short: it has a name, it exists, but for most real-world imaging uses an ordinary grayscale workflow or a higher-bit-depth pipeline is the more meaningful approach.