There is not much a jet engine could control.
There is no gearbox in a classical turbofan (not counting accessories such as generators). And those that have it (both future and present, and also turboprops) certainly don't have variable ratio: they barely cope with a single-stage box.
Stator blades do have adjustable pitch on many engines, but it is controlled according to the calculated flow velocity so that their angle was optimal for the conditions. It can't be used to help with rotor speed, or not much.
I'm not aware of any variable pitch rotors on turbojets/turbofans, except perhaps NK-93, which is almost a turboprop.
However, turboprops (and any prop engines) do (or can) use variable prop pitch for better response. Setting 'fine' pitch and thus higher RPM (for the same thrust) allows for faster response than the 'coarse' pitch setting. This is due to two factors: first, the prop is less 'loaded', and second, engine (particularly turbine) response is quicker near max RPM than near idle.
Supersonic engines on military aircraft can also use variable nozzle, but I'm not sure if dynamic response is intentionally included in their operating algorithm. Typically, the nozzle opens near idle, but this is rather for flow stability, esp. during start, than for RPM response.
So, fuel flow remains the main factor of thrust/RPM control. However, on all but the very first generation of turbojets, the pilot doesn't control the 'throttle' directly. (There is no 'throttle' as such). The thrust lever is just a signal to the engine controller, and the controller may modulate the fuel flow in a complex manner. Modern engines 'overcharge' the engine during spool-up, possibly injecting more fuel than in proportion to the lever, and even allowing higher peak temperature than normally allowed, all while controlling stability of the engine.