That's the first I've heard of it. Lacewings prey on aphids, and dragonflies are not considered major predators of mosquitoes. Again, it depends where you are. Anyway, numbers of mosquitoes or numbers of bites are not the main part of the story. The dominant factor is the survival rate of the insect.
Mosquitoes transmit diseases when they inject their saliva while they are feeding.
For a mosquito to transmit a virus or parasite that it has swallowed with its blood meal, it must itself be infected by these agents and they must eventually multiply in its salivary glands. All this has to happen before the insect dies. At summer temperatures, the process takes something like 10 days. If the mosquito dies before those 10 days are up obviously it won't transmit anything. This is where I really lose patience, because all these examples are carefully described in the scientific literature.