Birds are still around. Nature found a way in that specific case.
So if the extinction is swift and massive enough, the ecosystem will shift and there will not be a way to be found. So you should always ask: what dreadful bloodsucking beasts will replace the mosquito ?
I would much rather prefer they breed a transgenic mosquito unable to spread dengue than try to wipe all mosquitoes off the face of the planet.
Nature is just like a password cracker tries all possible combinations of characters. Most tries are wrong and the clock is always ticking.