Eradicating the diseases without eradicating the mosquitos is close to impossible, the vast majority of the diseases carried by mosquitoes require mosquitoes to complete their life cycle, such as malaria.
Secondly, diseases such as Yellow Fever rank among diseases such as HIV, and Chickenpox for being incurable and extraordinarily difficult to research treatments for, due to their structure and nature of attack.
Why start taking down a Jenga tower from the top? Wasting time and resources, when you could take it out from the bottom in one fell swoop.