Putting aside questions of ecosystems etc, I think the main reason is that we just can't - ironic since we seem to be extint-ing all the other animals
In South America they tried in the 50s and 60s, and more kept cropping up. They breed so quickly, if you miss an area they can just rebound. Then more can come in on ships and stuff
So you couldn't really localise it, it would have to be a huge global undertaking. And it would likely require widespread use of pesticides that are at best tricksy and at worst illegal, not to mention environmentally shitty
Most modern plans for eradication involve creating a virus that handles it, rather than a pesticide.
Have the virus introduce a gene that takes a few generations of breeding in the impacted population before it starts to debilitate or sterilize the mosquitoes. That way your virus can start to kill the population even as it spreads to areas that were missed.
Also significant politics within the field preventing integrated approaches to control. It's possible we could target specific species of mosquito that are vectors for deadly disease, with the intent of eradicating the disease by suppressing the vector. It would be the greatest collective undertaking of human kind. We'd have to shelf things like international borders and profits.
Eradication is really hard. If you just kill mosquitoes in a certain area, what's gonna stop them from coming back? You're just not gonna get all of them.
This way you introduce a mutation that can actually propagate through the gene pool, disabling the undesired trait for future generations. It's also highly selective, so that you don't accidentally get rid of other species or poison an area with pesticides.
Also living beings have no "purpose". They fill an evolutionary niche and shape the ecological system around them. The piss off us, so we play a little god, but nature has no opinions or morals whatsoever.
Well, they actually do have their ecological roles and it is always a hard decision if one should interfere on such a large scale with biological systems. We might think that we understand it, but it could be totally wrong. Really hard to predict. Mosquitoes are an important food source for other animals and are also pollinators.
Many species of mosquitos are reliant on blood for reproduction. The females utilize a "blood meal" for the nutrients for laying eggs to be fertilized. Additionally, it is the female mosquito bite that transmits diseases like malaria.