The problems eradication solves are obvious and important to solve.
The problems eradication might cause are less obvious and are potentially disastrous.
Many of the mosquitos are invasive species anyway. The point being, there are many many of species which would fill up the ecological niche left if we extincted a few.
Next to that it's not only the side effect of extinction of one species, but also the unforeseen side effects of the technologies used to accomplish this (extreme) extinction.
The whole point of technologists is that they really have to much faith that we both understand these complex ecosystems and complex technologies we just started using.
To try to put a technology on grand scale that you don't fully understand and can't control once in the wild is really stupid.