Not exactly. By having constant-Gm, a constant closed-loop BW is targeted. This keeps the transient response unchanged with common-mode / temperature. This relaxes the design for compensation as well as the timing requirements.
Constant open-loop gain is never a highly controllable spec and constant closed-loop gain is accurately achieved with precision passive components.
THD is a tricky spec here. If the input frequency is well withing UGB, then the loop-gain would be too high to suppress it's own variations, and if the frequency is close to UGB, the higher harmonics would be filtered automatically. Aol variation with ICMR would be available as well as Aol variation with frequency. You could work with these to get these numbers for your application.