Grip is not friction times weight
Where the schoolbook friction model breaks down on a tyre.
Grip is not simply friction multiplied by weight because the schoolbook friction model was built to describe a hard, unchanging block, and a tyre is soft rubber that changes shape under load, so pressing harder on it buys less and less extra grip for each extra bit of weight.
Rubber that moulds itself into the road
The schoolbook model of friction says that the sideways force needed to slide one surface across another is a fixed fraction of how hard the two surfaces are pressed together, and that fraction, the coefficient of friction, does not change however hard the pressing gets. That model was built from tests on rigid materials sliding across each other, where nothing physically deforms or interlocks, only the two flat surfaces rub, and it works well enough for a wooden crate dragged across a stone floor precisely because neither surface changes shape as it slides. A tyre's rubber does something the model never accounts for: it deforms into the microscopic texture of the road surface, flowing slightly into every tiny peak and valley of the tarmac the way nothing rigid could, and that deformation produces most of the sideways force a tyre can generate. A second, smaller contribution comes from the rubber's own internal friction as it is repeatedly stretched and released while rolling and sliding microscopically across the texture beneath it, a genuinely separate mechanism from the mechanical interlocking, but one that the schoolbook model is equally silent about, since it assumes a rigid slider that never internally deforms at all.
A fingertip dragged across wood
Press a fingertip onto a rough tabletop and drag it sideways, and the skin catches on the grain where a coin would glide. The soft pad of the finger squashes down into the tiny ridges of the wood, stretching and gripping into that texture as it drags, and the resistance felt comes from that stretching and interlocking. A tyre's rubber does the same thing at a much larger scale, moulding itself into the road's texture under load and generating grip from that moulding. This is also why a soft, sticky compound grips better than a hard one even at the same contact pressure, because the softer rubber deforms into the road's texture more completely, exactly as a soft fingertip conforms to rough wood better than a fingernail would. It is also why a warmed tyre grips better than a cold one of the identical compound, since warm rubber is softer and conforms more readily, the same reason a cold fingertip pressed onto rough wood feels stiffer and slides more easily than a warm one.
Each extra kilogram buys less grip
Because grip comes from this deformation, the sideways force a tyre can produce for each extra unit of weight pressing on it falls as the load rises, where the schoolbook model says it should stay constant. A lightly loaded racing tyre can generate a sideways force greater than the whole weight resting on it, since the rubber deforms easily and grips generously relative to that small load. Pile more weight onto the same tyre and the sideways force does keep rising, but not in step with the extra weight, because the contact patch is already deforming close to its limit and adding load mostly squashes the rubber flatter without producing a matching increase in grip. This is why extra weight never makes a car corner proportionally harder, and why spreading a car's weight across four tyres rather than concentrating it disproportionately on fewer of them tends to produce more total grip overall, since four tyres each working in their generous, lightly loaded range together produce more total sideways force than the same weight split unevenly, with some tyres pushed deep into their diminishing range while others sit barely loaded at all.
Lighter cars and balanced axles
A useful model of a tyre therefore has to be a curve that bends over as load increases, and a team setting up a car works from that curve instead of a single coefficient read off a specification sheet. It explains why reducing a car's weight helps cornering by more than the weight saving alone would suggest, since every kilogram removed is a kilogram that was generating grip at a worse rate than the kilograms already on the tyre. It also explains why keeping a car's weight balanced evenly between the axles tends to produce more total grip than concentrating the same total weight on one end, because each tyre then works nearer the lightly loaded, generously gripping part of its curve. Treating a tyre as a rigid block sliding on a fixed coefficient leads a design in the wrong direction, whether that means underestimating how much a lightweight car can corner relative to its weight, or overestimating how directly adding ballast in one corner of the car will translate into extra grip at that specific wheel.