There are 3,500 different kinds of mosquitos and only 100 of them bite humans. To the extent that mosquitos are food for something larger the mosquitos species that don't bite humans can do the job.
I'm not aware of research showing improved immune systems because of mosquito bites.
the mosquitos species that don't bite humans can do the job.
How large is the population of the problem species compared to the population of the benign species? There probably won't be more mosquitoes of the benign species if we eradicate the problem species, unless we start to farm the nice kinds of mosquitoes or something. Also the problem species might behave more predictably to certain predators than than the benign species, and populate certain niches that have certain predators that are used to have them around.
According to Louie Helm eradicating a species of mosquitoes could be done for as little as a few million dollar.
I don't have a few million dollar lying around so I can't spend my own money to do it. On the other hand, I think that on average every German citizen would be quite willing to pay 1€ per year to rid Germany of mosquitoes that bite humans.
That means it's a problem of public action. The German government should spend 80 million Euro to rid Germany of Mosquitos. That's an order of magnitude higher than the numbers quoted by Louie Helm).
The same goes basically for every country or state with mosquitos.
How could we get a government to do this without spending too much money ourselves? The straight forward way is writing a petition. We could host a website and simultaneously post a petition to every relevant parliament on earth.
How do we get attention for the petition? Facebook. People don't like Mosquitos and should be willing to file an internet petition to get rid of them. I would believe this to spread virally. The idea seems interesting enough to get journalists to write articles about it.
Bonus points:
After we have eradicated human biting mosquitoes from our homelands it's quite straightforward to export the technology to Africa.
Does anyone see any issues with that plan?