Short answer
Because near-neutral colours are made by mixing tiny amounts of pigment, and whichever pigment dominates becomes visible once the colour covers a large area. A beige with a red-leaning base reads as plainly beige on a card and as unmistakably pink across a whole wall. The same mechanism turns greys green, blue or violet.
A saturated colour has enough of one hue that small amounts of anything else are lost in it. A near-neutral does not — it is a balance between competing pigments, and anything that tips that balance shows. Two things tip it reliably: scale, because inter-reflection multiplies the dominant cast across the room, and light, because a light source deficient in part of the spectrum cannot render the pigments evenly. This is why the least colourful paints are the hardest to choose, which is the opposite of what most people expect.
Hold the candidate against plain printer paper. A beige that shows pink against white paper will show pink on a wall, only more so.
Casts are only visible in comparison. Three beiges side by side will separate into a pink one, a yellow one and a green one within seconds; any one of them alone looks simply beige.
Low-CRI LED and older fluorescent lighting render some parts of the spectrum poorly and can push a near-neutral strongly towards one cast after dark.
A beige that looks fine alone can turn pink beside a yellow-toned wood floor, because the eye judges it relative to the floor rather than in isolation.
Be particularly careful with near-neutrals in north-facing rooms and in rooms lit by low-CRI bulbs. Both conditions strip warmth unevenly, and a beige chosen elsewhere can read as pink, mauve or simply dirty once it is in place.
How to test it
Always test near-neutrals in threes, against printer paper, on more than one wall, in daylight and after dark. This is more testing than a strong colour needs, not less.
Undertone
This is the undertone problem in its purest form. The undertone of a near-neutral is not a subtlety to be aware of — it is the single property that determines whether the colour will work in your room.
Lighting
Two near-neutrals that match under daylight can look clearly different under artificial light, because they achieve the same appearance with different pigment mixes. That is metamerism, and near-neutrals are where it bites hardest in practice.
Finish
Sheen affects near-neutrals disproportionately, because there is little colour to look at and the eye falls on the reflection instead. A near-neutral in satin can look like a different colour from the same paint in matt.
Whatever a calculation or a chart says, view a physical sample in the actual location before committing to a large surface.
Colourwise editorial guidance, drawn from conventional decorating practice and from the physical behaviour of light and surfaces described in the linked lighting and finish guides. Not manufacturer advice — always follow the specific instructions supplied with any product you use.