Sterilize & release programs for feral cats work; the sterilized cats compete for the same resources as unsterilized ones, but are more effective and longer-lived because they don't have to support reproduction. This not only means that the sterilized cats won't reproduce, but the unsterilized ones will have a harder time, and reproduce less.
It seems likely that capturing cats, sterilizing them, and releasing them would have a larger impact than simply releasing a large number of sterile cats. It's also much easier to do in appreciable numbers, relative to the wild population, and it doesn't actually lead to extinction, just some additional population control. I don't think it is a good fit for mosquitos.
I once heard a modified version of the plan, where you release male mosquitos with a "driving Y chromisome" - that is, a Y chromisome with a mutation that leads to male offspring. Supposedly this should spread in a population even when but eventually leads to extinction.
I think I recall there being some argument as to why this was unlikely, but I'm not able to produce it. Certainly, every female mosquito has two X chromosomes that have avoided being driven out by the Y, and choice of mate would factor into that, and so in the population of female mosquitoes I would tentatively expect there to be evolutionary pressure there...
There is obviously the question of whether it would be sufficient, in time - the figures around time-frame were surprisingly short - but that's not a solid guarantee. Even if it just produced a substantial collapse for a couple years, it might help substantially in fighting malaria, though.
it has to be said - if non-sterile mosquitos are having a hard time surviving because sterile mosquitos drank up all the blood - we're probably going to suffer more than they will
It works better than trapping & killing.