Why don't we see diverging nozzles or diffusers? When it comes to fully subsonic propulsion, all outlets are converging nozzles (I did say fully subsonic so no converging diverging). From what I understand, they are flow regulating devices (changing the exit diameter allows to control the mass flow through the propulsion system, be it a turbofan or ducted electric fan).

Why don't we ever use diverging outlets, i.e. diffusers? Is it due to the back pressure? Wouldn't such a shape increase the pressure recovery, i.e. more thrust.
The basic answer is that pressure differential at the exit isn't what you want for thrust.  You want the exhaust stream to have velocity.  If it exits the nozzle with excess pressure, that extra pressure essentially gets converted into plume size (see the underexpanded streams on rocket first stages at high altitudes as an example).

This is why subsonic nozzles converge, and supersonic nozzles converge to a Mach 1 throat and then diverge.