That’s more something bat enthusiasts say to get people to like the critters. In reality insectivorous bats primarily eat larger, easier to catch prey like moths.

Given the small size of mosquitoes, restrictions imposed on prey detectability by low frequency echolocation, and variable field metabolic rates, mosquitoes may not be available to or profitable for all bats. [1]

[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3795000/

Sure, some do. But I don't think any bats eat only mosquitoes. As long as there are enough insects to eat, I think they'll be fine.