You can cut a dandelion off at ground level and it will be back within a fortnight. That is not stubbornness. It is architecture.
A plant built in two parts
A dandelion is a taproot with a crown on top and leaves arranged in a flat rosette. The root is thick, pale and often longer than the plant is tall.
That root is the reason the plant survives. It stores the sugar made by the leaves, and it can push out new growth when everything above ground is gone.
Why mowing does not work
The rosette grows flat, below the height of a mower blade. Only the flower stem rises, and cutting it removes flowers without touching the plant.
Worse, cutting the top off removes a brake. The crown holds dormant buds that stay quiet while the plant is growing normally, and wake up when the top is damaged.
Two ways to spread
Seed is the obvious one. A single flower head can produce well over a hundred seeds, each with a parachute that carries it on the wind for kilometres.
The second way is the root. Digging a dandelion out and leaving a fragment behind is effectively planting a new one.
Why a thin lawn loses
Dandelion seed needs light and bare soil to establish. A dense, tall lawn shades the soil surface and the seed fails.
This is why the answer to a weedy lawn is usually grass, not weedkiller: mow higher, feed in autumn, and overseed the thin patches so there is no bare ground to colonise.
Timing, in one line
In spring the root is full of reserves and regrowth is fast. In early autumn the plant is moving sugar down into the root for winter, which is the point at which it is most vulnerable.
If you are going to dig, dig then, and take the whole root with a fork rather than pulling the top off with your hand.
And a fair note: dandelions are among the first flowers to feed bees in spring. A lawn with a few is not a failure, and a lawn with none is not automatically healthier.
