There is a practical limit to the amount of compression that can be achieved in a single stage of an axial compressor. As a general principle, the rotor blades increase the velocity of the gas without much increase in pressure, and the stator blades increase the pressure while reducing the velocity.
The changes in the gas velocity are mainly in the circumferential direction. The maximum circumferential exit velocity of the gas that can be produced is related to the circumferential velocity of the rotor blades themselves (i.e. radius times angular velocity).
If you can design a blade with a short chord length which can do that (as in real-life compressors) there is no value in making the chord length of a single compressor stage longer. Even if you could get a higher compression ratio when the machine was running at one particular design condition of flow rate and gas input temperature and pressure, there would be a greater loss of performance and efficiency away from that design point.
The way you get a "longer chord length," to give a higher compression ratio, is by making a multi-stage compressor with alternating rotors and stators. Jet engine compressors may have 10 or more stages in total, on two or three different rotors running at different speeds.