They absolutely serve purpose in nature, they are a significant food source for bats and many other insects and males are pollinators.

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

They need blood to procreate so the method in the post does exactly what you are asking for

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.

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.

Bats eat mosquitoes so we be killing off bats food supply. So just get bats and solve your mosquitoe problem.