Is measuring noise as V/√Hz obsolete?
No, V/rtHz still matters.

I suspect the mistake you're making is to think that sampling at 1Hz reduces the bandwidth of the input signal. It doesn't, it just undersamples it. 

Now if your noise source is truly broadband, then the 20MHz limiter would reduce its amplitude. If it doesn't, that suggests the noise is band limited to something less than 20MHz. Which is consistent with my understanding of an avalanche effect Zener diode.

Make up a 1 kHz R-C filter and try again.