Random ENF would not work, since you could filter it out and look only at the 60 hz +- a few ppm signal. You need a random source that is very very close to 60 hz, but just slightly different.

But that might not be enough - the original signal is still there, and it's much stronger than your noise. The frequency is still powering your power supply, and still causing a 60 hz variation in the voltage.

You would need to isolate the power supply completely, or run it entirely through your noise adder. It would work best if you could add the noise to the DC output of the power supply.