Dragonflies are also an important prey item for birds and bats, both of which have decreased in numbers in recent years and would benefit from more food sources.
We can restore some of our wetlands, where dragonflies breed, and try to eliminate standing water in urban areas as much as possible -- old tires, ditches, etc. Mosquitoes love wetlands too, but in a properly balanced system, their populations would be controlled by fish, dragonflies, and amphibians.
We should also continue research into gene drives and similar efforts to reproductively target a species for extinction.
At the same time, we should be careful not to damage the thousands of non-biting mosquitoes which are valuable pollinators. About 5000 known species exist of which only about 100 feed on blood, and about 9 of those blood sucking species carry deadly diseases. If we target just the blood sucking species while not affecting the others, the effects on the ecosystem should be minimal.
Mosquitoes are a remarkably effective and robust creature, dating back 100 million years, and existing in nearly every part of the world. They bring great misery to many species of animals and their extinction will not be mourned.
The best option if you really want to wipe out mosquitoes is to invest the actual time and resources required into proving that mosquitoes can be targeted precisely, and that there won't be knock-on effects.
People get angry about that answer because that's not fast or convenient, and in the meantime people are dying of malaria. And I really do understand that, but there is no fast, easy, safe way to eliminate a species. Every possible solution takes time. I'm sure my boss gets upset too whenever I tell them that the new database upgrade needs to go through QA again.
The short answer is if you want to help people with malaria today and not just theoretically help them in the future, you should invest in building out Africa's infrastructure and increasing medical access and standards of living for the people who live there. That's also a pretty difficult task, but it may end up being faster, safer, and easier than eliminating mosquitoes.
Of course the outcome as is proves that males seem to make a poor long term solution since healthy ones can one to many enough to largely recover. It could theoretically weaken their genepool into nonviability if done persistently enough that the only mutant defective and inbred will survive but that would probably take a long time to converge to extinction.
Even if the number of certain birds or bats were somewhat reduced, I'd accept that tradeoff.