Yes, mosquitoes do pollinate and do feed larger animals.

But there are a lot of species of mosquitoes, even a lot of species that feed on humans. If we wipe out a particular species that feeds on humans, it will probably be replaced by another species that feeds on humans and the animals that eat them will adapt. Pollination is usually more specific.

Reducing annoyance means reducing the total population of mosquitoes that feed on humans, at least locally. That would have a bigger consequence for the species that feed on mosquitoes, though I don't know how big.

The main focus for many newer efforts to control mosquitos, especially the efforts focusing on genetically modified vectors, is on temporary local extinction. For many mosquito borne human diseases (malaria, dengue chikungunya....) we, humans, are the only known hosts. Killing all of the mosquitos in an area, allowing all sick/infected people to be treated/recover, then allowing mosquitos to repopulate the area would, in theory, end disease transmission. I don't know if we fully understand the ecologic impacts of something like this but, listening to the experts you hear everything from mosquitos don't matter to dragonflies will go extinct. Hard to parse out the truth.

Most mosquito species don't pray on humans. For anti-disease and anti-annoyance practice we would only need to kill those mosquito species that bite humans.

In additional humanity likely killed a few species with DDT that was used against mosquitos. I think it's plausible that the world loses less species if we would remove the mosquitos that bite humans.