Boeing 747 stabilizer trim is achieved by rotating the entire tailplane by up to 15 degrees. The elevator is much smaller (less than 1/3 the area) and is deflected by a smaller angle. That's some serious force you'd have to apply with those air jets either throughout the cruise phase or takeoff/landing.

Could even be "fuel trimming" between front and back wings; with the drawback that the action would be limited during both full and empty fuel states.

I'd assume there would be some constant air coming out of some of the nozzles for stability anyway.

Funnily that's applicable to balancing unswept flying wings, as all high lift/low drag airfoils are unstable in pitch (though many are stable for inverse flight, i.e. if you get scared you'd have to turn upside down and are suddenly stable).

I'm general, moving cog is potentially lossless, putting control surfaces into the airstream will always add drag.