Differential Amplifier Current Source

All your problems would come from the finite output impedance of the NMOS current source.

Having constant gain would be one problem. Since depending on the input, the tail current from the NMOS would vary and hence the gain would vary.
Input dependent gain variation would lead to non-linearity. This can be reduced to an extent if your circuit is in negative feedback.

An advantage of using NMOS tail current over an ideal current would be the fact that it is possible to make an NMOS transistor but impossible to make an ideal current source!

If CMRR or other issues arising from a real tail current is affecting the performance of the circuit, you need to look into better tail curent sources/sinks. Such as cascoded mirrors which would have much better output impedance which should definitely help. Bt they will come at the cost of headroom.

A current source with a significantly higher impedance is the 3-transistor Wilson circuit: