Well, perhaps not if the specific species that carry the diseases vanish. Eliminating all mosquitoes would almost certainly have an effect.
Of course, that's one reason why gene drives are so attractive -- you can target individual species, rather than indiscriminately killing every insect in the area.
Many other mosquitoes will just stick it out and hope you don't notice them before they fill up. They either succeed in one bite or get squashed. Those mosquitoes have less opportunity to acquire a disease from an infected human, and less opportunity to transmit one to an uninfected human. And if there is no mechanism for that vector to pass to a mosquito's offspring, the one-bite behavior means there is no transmission between humans at all.
The parasites don't survive equally well even within the species they inhabit.
Eliminating them would be devastating to Scandinavia's biodiversity.
What's the solution to diseases? Improving peoples' general health and wellbeing, so that their immune systems can handle the disease. The solution is not to eradicate an entire species in the middle of an already distraught global ecological catastrophe.
For every instance of a mosquito causing a malarial infection, there are likely uncountable instances of mosquitoes causing inoculation.