Most mosquito-borne illnesses are caused by a small subset of specific species of mosquito. There are only a few that cause malaria, for example. Past eradication methods, such as blanketing areas in insecticides, killed mosquitos indiscriminately, even the ones that don't carry diseases (as well as other insects). This had consequences on the rest of the food chain. Humans have tried to eradicate many species to the detriment of ecosystems. Whether that's wolves causing deer populations to explode, dingoes causing the same for kangaroos, or cane toads being brought in to control cane beetles. We just do not have a good track record of trying to eradicate species for our own benefit, because ecosystems are far more complex than we actually understand. For example, did you know that mosquitoes are pollinators?

A far better method for controlling mosquitos in cities would be to aid in the reintroduction, protection and support of their natural predators. Those would include bats, birds and spiders, as well as lizards and amphibians in some regions. Part of that would also include the eradication of invasive species that outcompete local species, but have differing diets.

Here's some links though:
https://cordis.europa.eu/article/id/449267-should-we-eradicate-mosquitoes
https://tepapa.govt.nz/discover-collections/read-watch-play/new-zealand-mosquito-census/mosquito-facts/can-we-just
https://www.newscientist.com/lastword/mg25233643-900-if-mosquitoes-were-eradicated-what-would-be-the-consequences/
https://www.ncbi.nlm.nih.gov/books/NBK585176/
https://pmc.ncbi.nlm.nih.gov/articles/PMC6378608/
https://www.ncbi.nlm.nih.gov/books/NBK585188/
https://www.ncbi.nlm.nih.gov/books/NBK585164/