Results shown below, together with the LTSpice .asc file you can use to play around with this yourself.
First attached image shows FFT for the rectified DC (green), reference voltage (red) and X-reg output (blue) for the designed-for 10 mA output (top) and a more punishing 100 mA (bottom).
Second image shows an LTSpice screengrab for the LT1086 with bypassed adj pin under comparable loading. Input voltage in blue, output in green. This is a reasonable approximation of a "good" IC regulator.
Last image shows a plot of the exported LTSpice FFT data for the X-reg and the LT1086-12V (Cin 1000uF, Cout 100uF) both at nominal currents of 10 mA. The LT1086-12V is a reasonable substitute for a generic LM7812, i.e. a "bad" IC regulator.
A typical op amp will have sufficient PSRR to mop of the residual noise from the bypassed LT1086. The fixed LT1086-12V, on the...