It is missing the symmetry probably because you didn't simulate long enough like in your post #14. Can you run longer simulation?
If the intended application is to build a sample and hold circuit, than you are going on a wrong path. Sample and hold circuits using OTAs are a switched capacitor circuits and then just by switching the caps in 2 phases usually, you define also the OTA input common mode. I think you should build a test bench that is closer to the application you have in mind. This cap-cap OTA you have right now is good for running .ac analysis for the S/H but not for transient simulations.