Digging out the roots is the only way without using questionable chemicals but is probably impossible with dandelions unless they are very small. Once you had dug all over you would need to reseed or turf the dug areas.

Insects love the flowers, especially dandelions, so I have decided to let most of my lawn become a wildflower meadow, otherwise it is a never-ending battle against nature.

I dont agree about having to returf? There is lots you can do, once the spring comes and they start growing mow every few days to keep the grass short and encourage it to grow side ways, this also weakens the weeds, feed and weed treat the lawn after scarifying as per instructions ( usually twice in the growing season). Keep your flowerbeds weeded and dig out all the way to the root. Any bear patches left in the lawn should fill out but if too big resow that area.

Where are you in that photo? Can't see any lawn! Phil