(I'm a layperson, too, for the record.)
"I know. But when you remove mosquitoes from the picture, aren't you putting selection pressure on the diseases to be airborne?"
Not necessarily that specific pressure. Except in carefully controlled laboratory situations, we can't specify the selection pressure being applied. There are too many potential pressures at work, and the mutation outcomes are too stochastic. At best we can force pressure in general. The outcome of that pressure might be entirely different from what we expect it to be.
Let's say we eradicate mosquitos. What other vectors of transmission does a virus like Zika have? What other hosts? It's possible the virus finds a new insect-borne transmission pathway: say, ticks instead of mosquitos. It's possible the virus 'focuses' (to use the term very very loosely) on other hosts, and effectively ceases to be a human concern. I'd wager that either of these outcomes is the more likely adaptation case than a leap to airborne transmission.
Evolving an entirely new means of infectious transmission seems to be a much rarer adaptation than adapting through other means (increased infectious potential; severity of infection; adaptation to new host types; etc.). It's popular in TV and movies to speak about a virus "going airborne," but in actual record, that's usually not what happens. Evolution doesn't have any agency or self-direction; it usually arrives at the 'laziest' and least costly alternative in response to imposed pressures. In this scenario, evolving airborne survivability and transmissibility is probably more costly than adapting to whatever enzyme prevents fleas and ticks from being carriers.
Source: http://evolution.berkeley.edu/evolibrary/news/141003_ebola