While they don't age like living organisms, their outer protein coating wears away due to temperature and UV light, so like you said they sort of "degrade".
We wouldn't have to destroy all mosquitos, though. Only a few species that are carriers of human pathogens. Once they are gone, other, benign mosquitos will occupy their spot, and will be eaten by the same animals as before.
Such a vague blanket statement sounds like an article of faith and not an actual informed opinion about the specific species of mosquitos, the animals that eat them, and the fungibility of their food sources. Correct me if I'm wrong though.
The mosquito species in question is native to northern Africa, and is an invasive species pretty much everywhere else in the world, having only arrived 300-400 years ago.